#72300898

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 9135 conjugacy classes of subgroups; only the 2997 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^10|c^2=gj,e^2=ghj,b^a=bg,c^a=cgh,c^b=chi,d^a=dhij,d^b=dghj,d^c=dj,e^a=egh,e^b=eghj,e^c=eh,f^b=fgij,f^c=fgij,f^d=fhij,f^e=fgij

Conjugacy classes & character table

112 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, 3, F.7*F.10);;
gap> SetPower(coll, 5, F.7*F.8*F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.7*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.8*F.9);;
gap> SetConjugate(coll, 4, 1, F.4*F.8*F.9*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.7*F.8*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.7*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.7*F.8*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.8);;
gap> SetConjugate(coll, 6, 2, F.6*F.7*F.9*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.7*F.9*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.8*F.9*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.7*F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);