#6814807

order	512
nilpotency class	3
isoclinism	isoclinic class of #6814805
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1786 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=i,c^2=ghi,e^2=h,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,h^b=hi,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, 6722003);