#6465965

order	512
nilpotency class	3
isoclinism	isoclinic class of #6444406
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 ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1828 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)(11,20)(12,17)(13,29)(14,18)(15,19)(16,23)(21,33)(22,34)(24,31)(25,27)(26,39)(28,32)(30,37)(35,38)(36,40);(2,5)(6,8)(9,10)(11,20)(12,25)(13,18)(14,29)(15,19)(16,24)(17,27)(21,35)(22,34)(23,31)(26,32)(28,39)(30,36)(33,38)(37,40);(9,11,15,22)(10,16)(12,21,28,36)(13,23,14,24)(17,30)(18,20)(19,31)(25,33)(26,37,27,38)(29,34)(32,35)(39,40);(9,12,14,27,15,28,13,26)(10,17,18,25,19,32,29,39)(11,21,24,38,22,36,23,37)(16,30,20,33,31,35,34,40);(1,3,4,7)(2,6,5,8)(11,22)(16,31)(20,34)(21,36)(23,24)(30,35)(33,40)(37,38)

Polycyclic presentation:

2^9|c^2=ghi,d^2=h,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,h^b=hi,e^c=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6373161);