#511995

order	512
nilpotency class	4
isoclinism	isoclinic class of #511755
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#578	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

932 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=ei,b^2=e,d^2=i,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fh,g^a=gi,h^a=hi,f^c=fh,h^c=hi,g^d=gi

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, 419191);