#503177

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

1142 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)(6,13)(7,17)(8,10)(9,11)(14,29)(15,32)(16,27)(18,28)(19,21)(20,23)(22,33)(24,25)(30,34)(31,35);(5,6,9,13,12,16,11,27)(7,18,20,29,22,30,21,35)(8,15,10,26,25,32,24,36)(14,19,28,33,31,23,34,17);(1,2,3,4)(5,7,8,23,11,21,24,33,12,22,25,19,9,20,10,17)(6,14,26,29,13,28,32,30,16,31,36,35,27,34,15,18);(5,8,11,24,12,25,9,10)(6,15,27,36,16,32,13,26)(7,19,21,17,22,23,20,33)(14,30,34,29,31,18,28,35)

Polycyclic presentation:

2^9|b^2=e,c^2=fh,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,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, 410373);