#72328573

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

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 9283 conjugacy classes of subgroups; only the 3041 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)(3,5)(4,7)(6,8)(9,10,12,14)(11,15,16,13)(17,18,20,22)(19,23,24,21)(25,26,29,30)(27,28,31,32);(2,3)(4,8)(17,18)(19,21)(20,22)(23,24)(27,31)(28,32);(25,27)(26,28)(29,31)(30,32);(2,3)(4,8)(11,16)(13,15)(19,24)(21,23)(26,30)(28,32);(1,3,5,2)(4,6,8,7)(9,10)(11,15)(12,14)(13,16)(17,18)(19,21)(20,22)(23,24)(25,28)(26,27)(29,32)(30,31);(1,4,5,8)(2,6,3,7)(9,11,12,16)(10,13,14,15)(17,19,20,24)(18,21,22,23)(25,27)(26,28)(29,31)(30,32)

Polycyclic presentation:

2^10|a^2=h,e^2=ghij,f^2=ij,b^a=bhi,c^b=cg,d^a=dhij,e^a=egij,e^b=ehij,e^d=ehij,f^a=fij,f^b=fhij,f^d=fij,f^e=fgi

Conjugacy classes & character table

118 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72328746
maximal normal subgroup of	#72328746
maximal quotient of	#72331142

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