#297066

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	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

662 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:

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

Polycyclic presentation:

2^9|a^2=eh,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, 204262);