#143525

order	512
nilpotency class	4
isoclinism	isoclinic class of #143522
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1999
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#532	D₄
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

356 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 68 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)(3,11)(4,9)(5,8)(6,10)(7,12)(13,16)(14,15)(17,18,22,28)(19,30,33,46)(20,34,37,45)(21,29,23,27)(24,40,43,31)(25,39,44,32)(26,36,41,38)(35,47,48,42);(1,3)(2,7)(4,13)(5,12)(6,14)(8,15)(9,11)(10,16)(17,19,22,33)(18,24,28,43)(20,35,37,48)(21,32,23,39)(25,26,44,41)(27,42,29,47)(30,45,46,34)(31,36,40,38);(2,8)(3,12)(4,5)(9,10)(11,15)(13,14)(17,19)(18,25)(20,31)(21,39)(22,33)(23,32)(24,41)(26,43)(27,46)(28,44)(29,30)(34,47)(35,38)(36,48)(37,40)(42,45)

Polycyclic presentation:

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

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, 50721);