#6916089

order	512
nilpotency class	3
isoclinism	isoclinic class of #6907993
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	#30714	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1478 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=f,c^2=fg,d^2=f,e^2=f,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,e^d=ei

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, 6823285);