#6971541

order	512
nilpotency class	3
isoclinism	isoclinic class of #6907869
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1364 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=gi,d^2=f,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=dfh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6878737);