#6814816

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

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

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,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, 6722012);