#72265724

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#15314
commutator, Frattini, cosocle	C₂⁴×C₄	C₂⁴
prime radical	C₂³×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂⁴×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂ ▷ C₁

5967 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

(2,4)(5,6,9,14,10,15,23,11)(7,12,17,25,18,28,24,32)(8,19,21,26,22,13,29,31)(16,33,20,30,34,27,36,35);(6,11)(9,23)(12,25)(13,26)(14,15)(16,34)(17,24)(19,31)(21,29)(27,30)(28,32)(33,35);(2,4)(5,7)(6,12,15,28)(8,20,22,36)(9,24)(10,18)(11,25,14,32)(13,30)(16,29,34,21)(17,23)(19,35)(26,27)(31,33);(1,2,3,4)(5,8)(6,13)(7,16,18,34)(9,21)(10,22)(11,26)(12,27,28,33)(14,31)(15,19)(17,20,24,36)(23,29)(25,35,32,30)

Polycyclic presentation:

2^10|a^2=e,c^2=fg,d^2=ghj,e^2=j,b^a=beg,c^a=cf,d^a=dh,d^b=dg,d^c=defhi,e^b=ej,e^c=ej,f^a=fj,f^c=fj,f^d=fj,g^c=gj,i^d=ij

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72265736, #72314069, #72314245
maximal normal subgroup of	#72265736, #72314069, #72314245

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