#6814835

order	512
nilpotency class	3
isoclinism	isoclinic class of #6814815
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 ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1786 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)(3,7)(4,6)(5,8)(9,10)(11,20)(12,17)(13,30)(14,18)(15,19)(16,23)(21,35)(22,36)(24,31)(25,29)(26,39)(27,34)(28,40)(32,37)(33,38);(1,3)(2,5)(4,8)(6,7)(9,10)(11,20)(12,25)(13,18)(14,30)(15,19)(16,24)(17,29)(21,35)(22,33)(23,31)(26,32)(27,40)(28,34)(36,38)(37,39);(9,11,15,21)(10,16)(12,26,27,22)(13,23,14,24)(17,33)(18,20)(19,31)(25,39)(28,38,29,37)(30,35)(32,34)(36,40);(2,6)(5,7)(11,21)(12,27)(16,31)(17,34)(20,35)(23,24)(25,40)(28,29);(3,8)(5,7)(9,12,14,29,15,27,13,28)(10,17,18,25,19,34,30,40)(11,22,24,38,21,26,23,37)(16,32,20,36,31,33,35,39)

Polycyclic presentation:

2^9|c^2=ghi,e^2=h,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,e^c=ei,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, 6722031);