#508938

order	512
nilpotency class	4
isoclinism	isoclinic class of #508304
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#493	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 806 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) = 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)(13,14,19,32,28,36,30,39,16,20,17,22,18,23,34,44)(15,24,41,37,31,43,29,21,25,42,26,40,27,38,33,35);(1,2,3,4)(6,9)(10,12)(14,20)(21,35)(22,32)(23,36)(24,42)(37,40)(38,43)(39,44);(2,4)(7,11)(10,12)(13,15)(14,21)(16,25)(17,29)(18,31)(19,33)(20,35)(22,38)(23,40)(24,39)(26,30)(27,28)(32,43)(34,41)(36,37)(42,44);(5,7,8,11)(6,10,9,12)(14,20)(15,25)(22,32)(23,36)(26,41)(27,31)(29,33)(39,44)

Polycyclic presentation:

2^9|a^2=fi,b^2=g,d^2=ei,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,d^c=de,f^c=fh,h^c=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, 416134);