#145516

order	512
nilpotency class	4
isoclinism	isoclinic class of #145508
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	#466
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 277 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)(3,10)(4,12)(5,13)(6,9)(7,11)(8,15)(14,16)(17,18,24,30)(19,31,34,44)(20,25,32,46)(21,39,22,41)(23,43,40,29)(26,37,47,33)(27,35,28,36)(38,48,42,45);(1,3)(2,7)(4,8)(5,14)(6,12)(9,16)(10,13)(11,15)(17,19,24,34)(18,25,30,46)(20,35,32,36)(21,38,22,42)(23,37,40,33)(26,27,47,28)(29,48,43,45)(31,41,44,39);(1,4,5,7,6,10,15,16)(2,3,8,11,9,12,13,14)(17,20,36,19,23,38,22,33,24,32,35,34,40,42,21,37)(18,26,39,48,29,31,28,25,30,47,41,45,43,44,27,46)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 52712);