#72337546

order	8192
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁸ · Schur multiplier (H₂) = C₂²⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#72266615
commutator, Frattini, cosocle	C₂⁵	C₂⁸
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

At least 26942 conjugacy classes of subgroups; only the 419284 normal subgroups are shown.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^13|d^c=di,e^b=ei,f^b=fj,f^d=fk,g^a=gj,g^e=gl,h^a=hk,h^c=hl,i^f=im,j^e=jm,k^c=km,l^a=lm

Conjugacy classes & character table

404 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72337555, #72338102
maximal normal subgroup of	#72337555, #72338102

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", "m");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetConjugate(coll, 4, 3, F.4*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.11);;
gap> SetConjugate(coll, 7, 1, F.7*F.10);;
gap> SetConjugate(coll, 7, 5, F.7*F.12);;
gap> SetConjugate(coll, 8, 1, F.8*F.11);;
gap> SetConjugate(coll, 8, 3, F.8*F.12);;
gap> SetConjugate(coll, 9, 6, F.9*F.13);;
gap> SetConjugate(coll, 10, 5, F.10*F.13);;
gap> SetConjugate(coll, 11, 3, F.11*F.13);;
gap> SetConjugate(coll, 12, 1, F.12*F.13);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);