#145290

order	512
nilpotency class	4
isoclinism	isoclinic class of #145268
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	C₂³	#482
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

248 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 64 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)(3,4)(5,14)(6,9)(7,10)(8,16)(11,13)(12,15)(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);(1,3,6,11)(2,7,9,12)(4,8,13,14)(5,15,16,10)(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);(1,4,5,12,6,13,16,7)(2,3,8,10,9,11,14,15)(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,b^2=g,c^2=d,d^2=g,e^2=hi,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fg,h^a=hi,c^b=cf,d^b=dg,e^b=eg,f^b=fi,e^c=egh,f^c=fgi,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, 52486);