#72300916

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹¹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁶
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

At least 8419 conjugacy classes of subgroups; only the 3029 normal subgroups are shown.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^10|a^2=g,b^a=bgi,c^a=cg,c^b=cg,d^a=dgh,d^b=dghj,d^c=dgj,e^a=egj,e^b=eghj,e^c=eghi,e^d=eghj,f^a=fg,f^b=fg,f^d=fgh

Conjugacy classes & character table

124 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72300929
maximal normal subgroup of	#72300929

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