#72306611

order	4096
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁸ · Schur multiplier (H₂) = C₂²⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁸
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1643471 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 8 generators on 32 points:

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

Polycyclic presentation:

2^12|c^b=ci,d^a=di,e^c=ej,f^a=fj,g^d=gk,g^f=gl,h^b=hk,h^e=hl

Conjugacy classes & character table

406 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307121, #72308529, #72321398, …
maximal normal subgroup of	#72307121, #72308529, #72321398, …

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", "l");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetConjugate(coll, 3, 2, F.3*F.9);;
gap> SetConjugate(coll, 4, 1, F.4*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.11);;
gap> SetConjugate(coll, 7, 6, F.7*F.12);;
gap> SetConjugate(coll, 8, 2, F.8*F.11);;
gap> SetConjugate(coll, 8, 5, F.8*F.12);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);