#502857

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
prime radical	C₂×#3176	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 903 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)(16,27)(17,28)(19,30)(21,36)(22,35)(24,29)(25,40)(26,39)(31,34)(32,41)(33,44)(37,38)(42,43);(1,2)(3,4)(6,9)(10,11)(13,14)(15,19)(16,21)(17,22)(18,20)(23,30)(24,31)(25,33)(26,32)(27,36)(28,35)(29,34)(37,42)(38,43)(39,41)(40,44);(2,4)(13,15,24,40,17,25,16,23,18,26,27,38,28,37,29,39)(14,19,31,44,22,33,21,30,20,32,36,43,35,42,34,41);(5,7,8,12)(6,10,9,11)(14,20)(19,32)(21,34)(22,35)(30,41)(31,36)(33,42)(43,44)

Polycyclic presentation:

2^9|c^2=fhi,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, 410053);