H(11,𝔽₂) #72258202

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂¹⁰ · Schur multiplier (H₂) = C₂⁴⁴

Aliases

extraspecial	2₊¹⁺¹⁰
Heisenberg	H(11,𝔽₂)

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂	C₂¹⁰
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ G
LECS	G ▷ C₂ ▷ C₁

229974158 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 10 generators on 64 points:

(2,8)(9,23)(10,24)(11,25)(12,26)(27,43)(28,44)(29,45)(30,46)(31,47)(32,48)(49,58)(50,59)(51,60)(52,61)(62,64);(1,2)(3,9)(4,10)(5,11)(6,12)(7,8)(13,23)(14,27)(15,28)(16,29)(17,24)(18,30)(19,31)(20,25)(21,32)(22,26)(33,43)(34,44)(35,45)(36,49)(37,50)(38,51)(39,46)(40,47)(41,52)(42,48)(53,58)(54,59)(55,60)(56,62)(57,61)(63,64);(3,13)(9,23)(14,33)(15,34)(16,35)(27,43)(28,44)(29,45)(36,53)(37,54)(38,55)(49,58)(50,59)(51,60)(56,63)(62,64);(1,3)(2,9)(4,14)(5,15)(6,16)(7,13)(8,23)(10,27)(11,28)(12,29)(17,33)(18,36)(19,37)(20,34)(21,38)(22,35)(24,43)(25,44)(26,45)(30,49)(31,50)(32,51)(39,53)(40,54)(41,56)(42,55)(46,58)(47,59)(48,60)(52,62)(57,63)(61,64);(4,17)(10,24)(14,33)(18,39)(19,40)(27,43)(30,46)(31,47)(36,53)(37,54)(41,57)(49,58)(50,59)(52,61)(56,63)(62,64);(1,4)(2,10)(3,14)(5,18)(6,19)(7,17)(8,24)(9,27)(11,30)(12,31)(13,33)(15,36)(16,37)(20,39)(21,41)(22,40)(23,43)(25,46)(26,47)(28,49)(29,50)(32,52)(34,53)(35,54)(38,56)(42,57)(44,58)(45,59)(48,61)(51,62)(55,63)(60,64);(5,20)(11,25)(15,34)(18,39)(21,42)(28,44)(30,46)(32,48)(36,53)(38,55)(41,57)(49,58)(51,60)(52,61)(56,63)(62,64);(1,5)(2,11)(3,15)(4,18)(6,21)(7,20)(8,25)(9,28)(10,30)(12,32)(13,34)(14,36)(16,38)(17,39)(19,41)(22,42)(23,44)(24,46)(26,48)(27,49)(29,51)(31,52)(33,53)(35,55)(37,56)(40,57)(43,58)(45,60)(47,61)(50,62)(54,63)(59,64);(6,22)(12,26)(16,35)(19,40)(21,42)(29,45)(31,47)(32,48)(37,54)(38,55)(41,57)(50,59)(51,60)(52,61)(56,63)(62,64);(1,6)(2,12)(3,16)(4,19)(5,21)(7,22)(8,26)(9,29)(10,31)(11,32)(13,35)(14,37)(15,38)(17,40)(18,41)(20,42)(23,45)(24,47)(25,48)(27,50)(28,51)(30,52)(33,54)(34,55)(36,56)(39,57)(43,59)(44,60)(46,61)(49,62)(53,63)(58,64)

Polycyclic presentation:

2^11|b^a=bk,d^c=dk,f^e=fk,h^g=hk,j^i=jk

Conjugacy classes & character table

1025 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	#72274291, C₂×H(11,𝔽₂)
maximal normal subgroup of	#72274291, C₂×H(11,𝔽₂)
maximal quotient of	C₂×H(11,𝔽₂)

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j", "k");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetConjugate(coll, 2, 1, F.2*F.11);;
gap> SetConjugate(coll, 4, 3, F.4*F.11);;
gap> SetConjugate(coll, 6, 5, F.6*F.11);;
gap> SetConjugate(coll, 8, 7, F.8*F.11);;
gap> SetConjugate(coll, 10, 9, F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);

H(11,𝔽2) #72258202