#72267566

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁰×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁵	C₂⁶
prime radical	C₂⁷	C₂⁴
LCS	G ▷ C₂⁵ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

133404 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 6 generators on 48 points:

(9,10)(11,12)(13,15)(14,16)(17,18)(19,20)(21,23)(22,24)(25,26)(27,29)(28,30)(31,32);(9,11)(10,12)(13,14)(15,16)(17,19)(18,20)(21,22)(23,24)(25,27)(26,29)(28,31)(30,32)(34,38)(39,46)(40,44)(47,48);(3,7)(6,8)(9,12)(10,14)(11,15)(13,16)(17,20,21,24)(18,22,23,19)(27,31)(29,32)(35,41)(36,43)(39,46)(40,44);(2,5)(6,8)(11,14)(12,16)(18,23)(19,22)(25,26,28,30)(27,29,31,32)(33,34,37,38)(35,39,41,46)(36,44,43,40)(42,48,45,47);(1,2)(3,6)(4,5)(7,8)(9,12)(10,11)(13,16)(14,15)(17,19)(18,20)(21,22)(23,24)(25,26)(27,29)(28,30)(31,32)(33,35,37,41)(34,39,38,46)(36,45,43,42)(40,48,44,47);(1,3)(2,6)(4,7)(5,8)(9,10)(11,12)(13,15)(14,16)(17,20)(18,19)(21,24)(22,23)(25,27)(26,29)(28,31)(30,32)(33,36,37,43)(34,40,38,44)(35,42,41,45)(39,47,46,48)

Polycyclic presentation:

2^11|c^2=hi,d^2=gik,e^2=k,f^2=k,c^a=ch,c^b=cg,d^a=dgh,d^b=dhk,d^c=di,e^c=ek,e^d=ej,f^c=fjk,f^d=fk,f^e=fk

Conjugacy classes & character table

188 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#72267571, #72267589
Fitting subgroup of	#72267571
maximal subgroup of	#72267568, #72267571
maximal normal subgroup of	#72267568, #72267571

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