#6516937

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500226
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3221 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,d^2=hi,g^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,g^b=gi,d^c=dh,f^c=fi,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, 6424133);