#72300920

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 8176 conjugacy classes of subgroups; only the 3037 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:

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

Polycyclic presentation:

2^10|a^2=h,b^2=g,c^2=ghj,b^a=bi,c^a=cgh,c^b=cgh,d^a=dghj,d^c=dghj,e^a=eh,e^b=eg,e^c=ei,e^d=eg,f^a=fh,f^b=fgj,f^c=fghj,f^d=fghj

Conjugacy classes & character table

130 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.8);;
gap> SetPower(coll, 2, F.7);;
gap> SetPower(coll, 3, F.7*F.8*F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.9);;
gap> SetConjugate(coll, 3, 1, F.3*F.7*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.7*F.8);;
gap> SetConjugate(coll, 4, 1, F.4*F.7*F.8*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.7*F.8*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.7);;
gap> SetConjugate(coll, 5, 3, F.5*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.7);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 6, 2, F.6*F.7*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.7*F.8*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.7*F.8*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);