#6811142

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

1782 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)(9,10,14,18)(11,17,23,16)(12,25,24,33)(13,27,29,19)(15,21,31,20)(22,32,38,34)(26,39,28,35)(30,36,40,37);(1,2)(3,6)(4,8)(5,7)(9,11,14,23)(10,16,18,17)(12,15,24,31)(13,28,29,26)(19,35,27,39)(20,33,21,25)(22,30,38,40)(32,37,34,36);(9,12,15,23,14,24,31,11)(10,17,21,25,18,16,20,33)(13,26,30,38,29,28,40,22)(19,34,37,35,27,32,36,39);(1,3,5,4)(2,6,7,8)(10,18)(16,17)(19,27)(20,21)(25,33)(32,34)(35,39)(36,37);(1,4,5,3)(2,6,7,8)(9,13,15,30,14,29,31,40)(10,19,21,37,18,27,20,36)(11,22,12,26,23,38,24,28)(16,32,33,39,17,34,25,35)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=h,d^2=fi,e^2=fhi,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, 6718338);