#146079

order	512
nilpotency class	4
isoclinism	isoclinic class of #144579
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#486
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#2071
prime radical	C₂³	#480
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1744 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

217 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 217 conjugacy classes of subgroups — too many to draw. Only its 70 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) = 128

A minimal faithful permutation representation — 3 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=d,d^2=g,f^2=hi,h^2=i,b^a=bd,c^a=ce,f^a=fg,c^b=cf,d^b=dg,e^b=eg,f^b=fh,h^b=hi,f^c=fgh,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, 53275);