#6811138

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

1758 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=h,d^2=fi,e^2=fh,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, 6718334);