#503123

order	512
nilpotency class	4
isoclinism	isoclinic class of #501060
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 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1272 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|b^2=e,c^2=fh,d^2=ei,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,h^a=hi,c^b=cg,d^b=di,f^b=fi,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, 410319);