#503160

order	512
nilpotency class	4
isoclinism	isoclinic class of #501244
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

1198 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 1198 conjugacy classes of subgroups — too many to draw. Only its 174 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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

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

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, 410356);