#297057

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

Aliases

No aliases for this group.

Subgroups

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

632 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=h,c^2=fh,d^2=ei,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, 204253);