#6684995

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682648
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

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

1947 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=fg,d^2=fi,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,h^c=hi,e^d=ei,h^e=hi

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, 6592191);