#6685500

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682628
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)(3,6)(4,5)(7,8)(9,10)(11,17)(12,24)(13,18)(14,19)(15,29)(16,20)(21,30)(22,31)(23,32)(25,34)(26,33)(27,35)(28,36)(37,40)(38,39);(2,5)(3,7)(9,11)(10,17)(12,13)(14,25)(15,22)(16,23)(18,24)(19,34)(20,32)(21,27)(26,38)(28,37)(29,31)(30,35)(33,39)(36,40);(9,12,13,23,16,22,15,11)(10,17,29,31,20,32,18,24)(14,25,26,38,28,37,27,21)(19,30,35,40,36,39,33,34);(2,5)(6,8)(9,13,16,15)(10,18,20,29)(11,12,23,22)(14,26,28,27)(17,24,32,31)(19,33,36,35)(21,25,38,37)(30,34,39,40);(1,3,4,7)(2,6,5,8)(9,14)(10,19)(11,21)(12,25)(13,26)(15,27)(16,28)(17,30)(18,33)(20,36)(22,37)(23,38)(24,34)(29,35)(31,40)(32,39)

Polycyclic presentation:

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

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, 6592696);