#72312848

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#15257
commutator, Frattini, cosocle	C₂⁴×C₄	C₂⁴
LCS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

4344 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^10|a^2=j,b^2=hj,c^2=h,f^2=j,b^a=bh,c^a=cf,c^b=cfh,d^a=de,d^b=deg,d^c=dgi,e^b=ei,e^c=ej,f^a=fj,f^b=fj,f^c=fj,g^a=gj,g^b=gj,g^c=gij,h^d=hij

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72312857, #72312882, #72313478
maximal normal subgroup of	#72312857, #72312882, #72313478

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