#6433241

order	512
nilpotency class	3
isoclinism	isoclinic class of #6433077
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	C₂×#537	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1944 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)(3,5)(4,6)(7,8)(9,10,15,20)(11,21,24,29)(12,25,28,17)(13,18,16,31)(14,23,30,19)(22,36,38,35)(26,39,32,33)(27,34,40,37);(1,3)(2,5)(4,7)(6,8)(9,11,14,18,15,24,30,31)(10,16,19,21,20,13,23,29)(12,22,27,33,28,38,40,39)(17,32,34,35,25,26,37,36);(3,7)(5,8)(9,12,14,27,15,28,30,40)(10,17,19,34,20,25,23,37)(11,22,18,33,24,38,31,39)(13,26,29,36,16,32,21,35);(2,6)(5,8)(9,13)(10,18)(11,23)(12,26)(14,29)(15,16)(17,33)(19,24)(20,31)(21,30)(22,37)(25,39)(27,36)(28,32)(34,38)(35,40);(11,24)(13,16)(18,31)(21,29)(22,38)(26,32)(33,39)(35,36)

Polycyclic presentation:

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

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, 6340437);