#6814812

order	512
nilpotency class	3
isoclinism	isoclinic class of #6814811
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

1805 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)(9,10,15,21)(11,16,24,33)(12,25,27,37)(13,18,30,23)(14,20,31,19)(17,34,36,29)(22,35,38,28)(26,40,39,32);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16)(12,26)(13,20)(14,23)(15,24)(17,35)(18,31)(19,30)(21,33)(22,29)(25,40)(27,39)(28,36)(32,37)(34,38);(9,12,15,27)(10,17)(11,22)(13,28,30,35)(14,29,31,34)(16,32,33,40)(18,26)(19,25)(20,37)(21,36)(23,39)(24,38);(2,6)(5,7)(12,27)(17,36)(22,38)(25,37)(26,39)(28,35)(29,34)(32,40);(3,8)(5,7)(9,13,14,16,15,30,31,33)(10,18,20,24,21,23,19,11)(12,28,29,32,27,35,34,40)(17,26,37,38,36,39,25,22)

Polycyclic presentation:

2^9|a^2=i,c^2=gh,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

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, 6722008);