#6806826

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806815
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

1949 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)(11,16)(12,24)(13,17)(14,18)(15,19)(20,30)(21,31)(22,25)(23,32)(26,37)(27,33)(28,34)(29,35)(36,40)(38,39);(1,3)(2,5)(4,8)(6,7)(9,11,15,23)(10,16,19,32)(12,25,17,30)(13,20,24,22)(14,28,29,36)(18,34,35,40)(21,37,38,27)(26,39,33,31);(9,12,13,10,15,17,24,19)(11,16,22,25,23,32,20,30)(14,26,27,18,29,33,37,35)(21,31,28,34,38,39,36,40);(2,6)(5,7)(9,13,15,24)(10,17,19,12)(11,20,23,22)(14,27,29,37)(16,30,32,25)(18,33,35,26)(21,36,38,28)(31,40,39,34);(3,8)(5,7)(9,14,24,37,15,29,13,27)(10,18,12,26,19,35,17,33)(11,21,20,36,23,38,22,28)(16,31,30,40,32,39,25,34)

Polycyclic presentation:

2^9|b^2=i,c^2=h,d^2=i,e^2=hi,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,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6714022);