#6808470

order	512
nilpotency class	3
isoclinism	isoclinic class of #6807031
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

1719 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|b^2=i,c^2=hi,d^2=fi,e^2=hi,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,g^c=gi,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, 6715666);