#295129

order	512
nilpotency class	3
isoclinism	isoclinic class of #268139
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#496	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

759 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2)(3,7)(4,5)(6,8)(9,10)(11,16)(12,22)(13,17)(14,19)(15,20)(18,23)(21,24)(25,26)(27,31)(28,36)(29,37)(30,32)(33,34)(35,40)(38,39);(2,5)(6,7)(9,11,13,20)(10,15,17,16)(12,14,21,23)(18,24,19,22)(25,27,30,35)(26,31,32,40)(28,33,37,39)(29,38,36,34);(9,12,23,20,13,21,14,11)(10,16,19,24,17,15,18,22)(25,28,36,26,30,37,29,32)(27,33,38,40,35,39,34,31);(1,3,4,8)(2,6,5,7)(10,17)(15,16)(18,19)(22,24)

Polycyclic presentation:

2^9|b^2=h,c^2=fh,d^2=ehi,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=ghi,g^c=gh

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 202325);