#145491

order	512
nilpotency class	4
isoclinism	isoclinic class of #145476
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	C₂³	#459
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

263 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 263 conjugacy classes of subgroups — too many to draw. Only its 72 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)(3,6)(4,7)(5,9)(8,13)(10,14)(11,16)(12,15)(17,18,24,30)(19,31,35,44)(20,25,21,38)(22,28,36,27)(23,40,37,29)(26,39,43,33)(32,42,46,45)(34,48,47,41);(1,3,5,11)(2,6,9,16)(4,12,13,10)(7,15,8,14)(18,25)(19,32)(20,36)(21,22)(23,37)(26,42)(27,29)(28,40)(30,38)(31,44)(33,47)(34,39)(35,46)(43,45);(2,7)(4,13)(6,14)(8,9)(10,12)(15,16)(17,19,20,33,37,47,22,32,24,35,21,39,23,34,36,46)(18,26,27,31,40,45,38,48,30,43,28,44,29,42,25,41)

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=d,d^2=gi,e^2=hi,f^2=g,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=fgi,g^b=gi,h^b=hi,e^c=egh

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 52687);