#143146

order	512
nilpotency class	4
isoclinism	isoclinic class of #143142
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#493
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#2011
prime radical	C₂²×D₈	D₄
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1744 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

435 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 435 conjugacy classes of subgroups — too many to draw. Only its 74 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)(3,9)(4,12)(5,8)(6,10)(7,14)(11,16)(13,15)(17,18,22,27)(19,28,31,46)(20,24,35,42)(21,36,34,26)(23,38,40,29)(25,43,41,33)(30,47,37,44)(32,45,48,39);(1,3)(2,6)(4,13)(5,11)(7,16)(8,15)(9,12)(10,14)(17,19,22,31)(18,23,27,40)(20,32,35,48)(21,37,34,30)(24,28,42,46)(25,44,41,47)(26,45,36,39)(29,43,38,33);(6,15)(9,16)(18,24)(20,33)(21,34)(23,28)(25,26)(27,42)(29,32)(30,37)(35,43)(36,41)(38,48)(39,47)(40,46)(44,45)

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=gi,e^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fghi,g^a=gi,h^a=hi,c^b=cf,d^b=dg,e^b=eg,g^b=gi,h^b=hi,d^c=dh,e^c=eh,g^c=gi,h^c=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 50342);