#508307

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

Polycyclic presentation:

2^9|a^2=f,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,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, 415503);