#145485

order	512
nilpotency class	4
isoclinism	isoclinic class of #145472
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2054
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#3169	C₂²
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×M₁₆ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 264 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)(3,6)(4,7)(5,9)(8,13)(10,14)(11,16)(12,15)(17,18)(19,30)(20,24)(21,37)(22,39)(23,29)(25,32)(26,35)(27,38)(28,36)(31,43)(33,40)(34,41)(42,47)(44,46)(45,48);(1,3,5,11)(2,6,9,16)(4,12,13,10)(7,15,8,14)(18,24)(19,31)(20,35)(21,38)(22,36)(26,28)(27,39)(29,37)(30,44)(32,34)(33,48)(40,41)(42,45)(46,47);(2,7)(4,13)(6,14)(8,9)(10,12)(15,16)(17,19,20,31,36,46,38,48,23,34,21,32,22,33,35,47)(18,25,26,41,39,45,37,44,29,43,27,30,28,42,24,40)

Polycyclic presentation:

2^9|b^2=ghi,c^2=d,d^2=gi,e^2=hi,f^2=gi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fh,g^a=gi,h^a=hi,c^b=cf,d^b=dg,e^b=eh,f^b=fg,g^b=gi,h^b=hi,e^c=egh,f^c=fi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 52681);