#503174

order	512
nilpotency class	4
isoclinism	isoclinic class of #500730
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#443 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1126 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:

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

Polycyclic presentation:

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

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, 410370);