#6810046

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 ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1996 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|c^2=fh,d^2=i,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, 6717242);