#502855

order	512
nilpotency class	4
isoclinism	isoclinic class of #501064
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

953 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 953 conjugacy classes of subgroups — too many to draw. Only its 166 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) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 44 points:

(5,6)(7,11)(8,9)(10,12)(15,23)(16,24)(17,27)(19,29)(21,30)(22,34)(25,39)(26,37)(28,38)(31,40)(32,44)(33,43)(35,36)(41,42);(1,2)(3,4)(6,9)(10,11)(13,14)(15,19)(16,21)(17,22)(18,20)(23,29)(24,30)(25,32)(26,31)(27,34)(28,33)(35,41)(36,42)(37,40)(38,43)(39,44);(2,4)(13,15,24,36,17,25,28,37,18,26,38,39,27,35,16,23)(14,19,30,42,22,32,33,40,20,31,43,44,34,41,21,29);(5,7,8,12)(6,10,9,11)(14,20)(19,31)(21,33)(22,34)(29,40)(30,43)(32,41)(42,44)

Polycyclic presentation:

2^9|c^2=fh,d^2=e,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=cg,d^b=di

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 410051);