#6815118

order	512
nilpotency class	3
isoclinism	isoclinic class of #6814827
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 ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1883 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=gh,d^2=f,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,d^c=di,e^c=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, 6722314);