#508300

order	512
nilpotency class	4
isoclinism	isoclinic class of #508298
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

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 ◁ C₂²×#41 ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

768 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 768 conjugacy classes of subgroups — too many to draw. Only its 146 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)(13,14,18,24,16,22,31,38,17,23,32,39,19,25,33,21)(15,26,29,42,30,36,44,35,28,20,34,37,27,40,43,41);(1,2)(3,4)(6,9)(10,12)(13,15)(14,20)(16,30)(17,28)(18,29)(19,27)(21,35)(22,40)(23,26)(24,37)(25,36)(31,44)(32,34)(33,43)(38,41)(39,42);(2,4)(5,7)(6,10)(8,11)(9,12)(14,21)(16,19)(18,33)(20,35)(22,39)(23,38)(24,25)(26,41)(27,30)(29,43)(31,32)(34,44)(36,37)(40,42);(7,11)(10,12)(13,16,17,19)(14,22,23,25)(15,27,28,30)(18,31,32,33)(20,36,26,40)(21,24,38,39)(29,43,34,44)(35,42,41,37)

Polycyclic presentation:

2^9|a^2=f,d^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,c^b=cg,d^b=di,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, 415496);