#145310

order	512
nilpotency class	4
isoclinism	isoclinic class of #145308
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2033
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

256 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 60 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)(3,7)(4,6)(5,9)(8,12)(10,13)(11,14)(15,16)(17,18,22,28)(19,23,32,38)(20,24,35,41)(21,26,34,27)(25,42,40,33)(29,43,31,37)(30,39,45,48)(36,46,47,44);(2,6)(3,10)(4,12)(7,14)(8,9)(11,16)(17,19)(18,23)(20,30)(21,29)(22,32)(24,39)(25,36)(26,43)(27,37)(28,38)(31,34)(33,44)(35,45)(40,47)(41,48)(42,46);(1,3,4,10,5,11,12,16)(2,7,8,15,9,14,6,13)(18,24)(19,29)(20,33)(21,34)(23,36)(25,27)(26,40)(28,41)(30,45)(31,32)(35,42)(37,48)(38,47)(39,43)

Polycyclic presentation:

2^9|a^2=i,c^2=d,d^2=g,e^2=hi,f^2=gi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,c^b=cf,d^b=dg,f^b=fgi,e^c=egh,f^c=fi,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, 52506);