#146252

order	512
nilpotency class	4
isoclinism	isoclinic class of #144122
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#493
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#2093
prime radical	C₂³	#497
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#322 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

230 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 84 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=d,b^2=deg,c^2=g,d^2=g,e^2=gi,f^2=hi,h^2=i,b^a=bd,c^a=ce,f^a=fi,c^b=cf,d^b=dg,e^b=eg,f^b=fh,h^b=hi,d^c=dgi,e^c=egi,f^c=fh,h^c=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53448);