#146135

order	512
nilpotency class	4
isoclinism	isoclinic class of #146135
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

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

normal subgroups only

This group has 186 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) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53331);