#72231900

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄×C₈ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	C₂³×C₄²
commutator	C₂×C₄	C₂³×C₄×C₈
Frattini, cosocle	C₈²	C₂⁵
socle	C₂²	C₂³×#322
prime radical	#3444	C₄²
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₄² ◁ G
LECS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁

4446 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 64 points:

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

Polycyclic presentation:

2^11|a^2=f,e^2=g,f^2=h,g^2=i,h^2=j,i^2=k,c^b=cj,d^b=dk,e^a=ei,f^e=fk,g^a=gk

Conjugacy classes & character table

416 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72231918, #72231972, #72322318
maximal normal subgroup of	#72231918, #72231972, #72322318

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> SetPower(coll, 1, F.6);;
gap> SetPower(coll, 5, F.7);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.9);;
gap> SetPower(coll, 8, F.10);;
gap> SetPower(coll, 9, F.11);;
gap> SetConjugate(coll, 3, 2, F.3*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.11);;
gap> SetConjugate(coll, 5, 1, F.5*F.9);;
gap> SetConjugate(coll, 6, 5, F.6*F.11);;
gap> SetConjugate(coll, 7, 1, F.7*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);