#72300908

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 8454 conjugacy classes of subgroups; only the 2981 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:

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

Polycyclic presentation:

2^10|a^2=h,b^2=g,c^2=ghij,e^2=g,b^a=bi,c^a=cgij,c^b=chij,d^c=dij,e^a=ej,e^c=ehij,e^d=eg,f^b=fghj,f^c=fghj,f^d=fh,f^e=fghj

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