#291414

order	512
nilpotency class	3
isoclinism	isoclinic class of #269211
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

751 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)(4,6)(5,8)(9,10)(11,17)(12,21)(13,16)(14,20)(15,19)(18,23)(22,24)(25,26)(27,31)(28,33)(29,32)(30,37)(34,38)(35,39)(36,40);(1,3,4,8)(2,5,6,7)(11,18)(12,19)(15,21)(17,23)(25,27,29,36)(26,31,32,40)(28,35,30,34)(33,39,37,38);(9,11,19,24,13,18,12,20)(10,14,21,23,16,22,15,17)(26,32)(27,34,36,35)(31,38,40,39)(33,37);(2,6)(5,7)(11,18)(14,22)(17,23)(20,24)(25,28,29,30)(26,33,32,37)(27,35,36,34)(31,39,40,38)

Polycyclic presentation:

2^9|b^2=ehi,c^2=fi,d^2=hi,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,d^c=dh,g^c=gh

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 198610);