#503163

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

1130 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 1130 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)(5,6,12,17)(7,13,22,18)(8,10,14,24)(9,15,27,25)(11,29,26,16)(19,23,28,33)(20,35,31,32)(21,36,30,34);(1,2,4,3)(5,7)(6,13)(8,23)(9,20)(10,28)(11,30)(12,22)(14,33)(15,35)(16,36)(17,18)(19,24)(21,26)(25,32)(27,31)(29,34);(5,8,24,6,11,25,9,16,12,14,10,17,26,15,27,29)(7,18,20,23,21,36,19,35,22,13,31,33,30,34,28,32);(1,3,4,2)

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=fhi,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, 410359);