#6464102

order	512
nilpotency class	3
isoclinism	isoclinic class of #6443052
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×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1732 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=g,c^2=g,d^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=dh,f^b=fi,e^c=ei,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, 6371298);