#72267741

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂⁶×C₄⁴

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

132540 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:

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

Polycyclic presentation:

2^11|c^2=gh,d^2=gj,e^2=k,f^2=k,c^a=ci,c^b=cg,d^a=dghij,d^b=dgik,d^c=djk,e^d=ejk,f^c=fjk,f^e=fk

Conjugacy classes & character table

212 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#72267748, #72267771
Fitting subgroup of	#72267748
maximal subgroup of	#72267743, #72267748
maximal normal subgroup of	#72267743, #72267748

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