#508165

order	512
nilpotency class	4
isoclinism	isoclinic class of #508008
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

890 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 890 conjugacy classes of subgroups — too many to draw. Only its 156 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)(7,10)(8,9)(11,12)(15,23)(16,26)(17,28)(19,31)(21,36)(22,35)(24,41)(25,40)(27,30)(29,39)(32,43)(33,44)(34,42)(37,38);(1,2)(3,4)(6,9)(10,12)(13,14,18,20)(15,19,25,32)(16,21,29,34)(17,22,28,35)(23,31,40,43)(24,33,30,38)(26,36,39,42)(27,37,41,44);(2,4)(7,11)(10,12)(13,15,18,25)(14,19,20,32)(16,27,29,41)(17,30,28,24)(21,37,34,44)(22,38,35,33)(23,39,40,26)(31,42,43,36);(5,7,8,11)(6,10,9,12)(14,20)(19,32)(21,34)(22,35)(31,43)(33,38)(36,42)(37,44)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 415361);