#507564

order	512
nilpotency class	4
isoclinism	isoclinic class of #499897
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #376 ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

529 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 529 conjugacy classes of subgroups — too many to draw. Only its 152 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) = 48
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2)(3,7)(4,5)(6,8)(17,18,22,27)(19,23,28,36)(20,24,33,40)(21,34,32,26)(25,41,39,31)(29,35,42,46)(30,45,43,38)(37,48,47,44);(2,5)(6,7)(9,10,12,14)(11,15,16,13)(17,19)(18,23)(20,29)(21,30)(22,28)(24,35)(25,37)(26,38)(27,36)(31,44)(32,43)(33,42)(34,45)(39,47)(40,46)(41,48);(9,11,12,16)(10,13,14,15)(18,24)(20,31)(21,32)(23,35)(25,26)(27,40)(29,44)(30,43)(33,41)(34,39)(36,46)(37,38)(42,48)(45,47);(1,3,4,8)(2,6,5,7)(19,28)(23,36)(29,42)(30,43)(35,46)(37,47)(38,45)(44,48)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=g,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,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, 414760);