#503131

order	512
nilpotency class	4
isoclinism	isoclinic class of #501076
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

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

1264 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 410327);