#143584

order	512
nilpotency class	4
isoclinism	isoclinic class of #143530
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	#3169	C₂²
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#446 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 353 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:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=g,d^2=gi,e^2=h,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,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=fhi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 50780);