#6834163

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806903
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#125 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1177 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=i,c^2=fhi,d^2=fi,e^2=gh,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6741359);