#72262688

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁷

Aliases

No aliases for this group.

Subgroups

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

At least 16835 conjugacy classes of subgroups; only the 29712 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 7 generators on 32 points:

(1,2)(3,6)(4,5)(7,8)(25,26,30,32)(27,31,29,28);(3,7)(6,8)(9,10)(11,13)(12,14)(15,16)(25,27)(26,31)(28,32)(29,30);(9,11)(10,13)(12,16)(14,15)(17,18)(19,20)(21,23)(22,24)(25,28)(26,27)(29,32)(30,31);(2,5)(6,8)(9,10)(11,15)(12,14)(13,16)(17,19)(18,22)(20,23)(21,24)(25,27)(26,31)(28,32)(29,30);(10,14)(13,15)(19,24)(20,22)(25,28)(26,27)(29,32)(30,31);(1,2)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,20)(18,19)(21,22)(23,24)(25,29,30,27)(26,28,32,31);(1,3)(2,6)(4,7)(5,8)(9,11)(10,15)(12,16)(13,14)(17,18)(19,22)(20,24)(21,23)(25,28)(26,29)(27,32)(30,31)

Polycyclic presentation:

2^11|a^2=i,f^2=i,c^b=ci,d^a=dh,d^c=dk,e^b=ehij,e^d=ek,f^b=fi,f^d=fjk,f^e=fik,g^a=gi,g^b=gj,g^f=gk

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72262698, #72262721, #72262761
maximal normal subgroup of	#72262698, #72262721, #72262761

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