#503175

order	512
nilpotency class	4
isoclinism	isoclinic class of #500733
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

1137 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)(7,20)(8,12)(9,18)(10,28)(11,15)(13,19)(14,22)(16,29)(17,27)(21,26)(23,32)(24,34)(25,33)(30,36)(31,35);(5,7,10,22,13,25,29,24)(6,14,16,20,19,34,28,33)(8,26,17,35,15,32,18,36)(9,21,11,30,27,23,12,31);(1,2,3,4)(5,8,27,28,10,17,12,6,13,15,9,16,29,18,11,19)(7,14,30,35,24,33,21,26,25,34,31,36,22,20,23,32);(5,9,10,11,13,27,29,12)(6,15,16,18,19,8,28,17)(7,21,22,30,25,23,24,31)(14,32,20,36,34,26,33,35)

Polycyclic presentation:

2^9|b^2=e,c^2=fh,d^2=e,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,d^c=di

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 410371);