#143473

order	512
nilpotency class	4
isoclinism	isoclinic class of #143452
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2011
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

349 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 349 conjugacy classes of subgroups — too many to draw. Only its 70 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=i,d^2=gi,e^2=gh,f^2=g,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egh,f^a=fgi,g^a=gi,h^a=hi,c^b=cf,d^b=dg,e^b=eghi,f^b=fgi,g^b=gi,h^b=hi,d^c=dh,e^c=egh,f^c=fgi,g^c=gi,h^c=hi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 50669);