#146133

order	512
nilpotency class	4
isoclinism	isoclinic class of #146133
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2077
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#532	D₄
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

190 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 190 conjugacy classes of subgroups — too many to draw. Only its 54 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:

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

Polycyclic presentation:

2^9|a^2=eh,c^2=d,d^2=g,e^2=h,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,f^a=fgi,c^b=cf,d^b=dg,e^b=egi,f^b=fi,e^c=eh,f^c=fgi,h^c=hi

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53329);