#72234751

order	4096
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#72217875
commutator, Frattini, cosocle	C₂⁶	C₂⁶
LCS	G ▷ C₂⁶ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁶ ▷ C₂² ▷ C₁

At least 24267 conjugacy classes of subgroups; only the 5279 normal subgroups are shown.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^12|b^2=h,f^2=k,b^a=bh,e^b=ei,e^c=ei,e^d=eh,f^a=fj,f^b=fjk,f^e=fg,g^a=gl,g^c=gl,g^d=gk,h^f=hk,i^f=il,j^b=jk,j^e=jl

Conjugacy classes & character table

214 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72238380, #72262293, #72305331, …
maximal normal subgroup of	#72238380, #72262293, #72305331, …
automorphism group of	#1462, #1463

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