#503165

order	512
nilpotency class	4
isoclinism	isoclinic class of #501248
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 ◁ #2931 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1175 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)(7,17)(8,19)(9,25)(10,24)(12,27)(13,14)(15,26)(18,31)(20,32)(21,22)(23,30)(28,36)(29,35)(33,34);(1,2,4,3)(5,6)(7,18)(8,23)(9,13)(10,26)(11,16)(12,21)(14,25)(15,24)(17,31)(19,30)(20,35)(22,27)(28,34)(29,32)(33,36);(5,7,19,34,10,22,25,32,11,20,9,21,24,33,8,17)(6,12,14,18,15,29,30,36,16,28,23,35,26,31,13,27);(1,3,4,2)(7,20)(12,28)(17,32)(18,35)(21,34)(22,33)(27,36)(29,31)

Polycyclic presentation:

2^9|b^2=e,c^2=fh,d^2=e,f^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=ghi,h^a=hi,c^b=cg,f^b=fhi,g^b=ghi,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, 410361);