#6808132

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806827
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

1755 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)(11,22)(12,27)(13,31)(14,19)(15,33)(16,21)(17,25)(18,30)(20,28)(23,40)(24,35)(26,36)(29,39)(32,38)(34,37);(1,2)(3,6)(4,5)(7,8)(9,11)(10,17)(12,28)(13,29)(14,25)(15,26)(16,34)(18,38)(19,22)(20,39)(21,36)(23,30)(24,32)(27,31)(33,37)(35,40);(9,12,16,30,14,29,15,24)(10,18,21,39,19,35,33,27)(11,23,26,28,25,32,34,13)(17,20,37,38,22,31,36,40);(1,3,4,7)(2,6,5,8)(10,19)(17,22)(18,35)(20,31)(21,33)(27,39)(36,37)(38,40);(2,5)(6,8)(9,13,15,32,14,28,16,23)(10,20,33,40,19,31,21,38)(11,24,34,29,25,30,26,12)(17,35,36,39,22,18,37,27)

Polycyclic presentation:

2^9|c^2=h,d^2=fi,e^2=hi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=ch,e^b=efh,h^b=hi,g^c=gi

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, 6715328);