#6463749

order	512
nilpotency class	3
isoclinism	isoclinic class of #6442587
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1862 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=g,d^2=h,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,f^b=fi,e^d=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, 6370945);