#143529

order	512
nilpotency class	4
isoclinism	isoclinic class of #143528
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	#19189	C₂
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

357 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 357 conjugacy classes of subgroups — too many to draw. Only its 64 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)(19,28)(20,34)(21,26)(22,40)(23,27)(24,30)(25,38)(29,35)(31,46)(32,43)(33,47)(36,37)(39,44)(41,48)(42,45);(1,3)(2,7)(4,13)(5,12)(6,14)(8,15)(9,11)(10,16)(17,19,23,31)(18,24,27,41)(20,35,39,45)(21,33,22,32)(25,43,36,47)(26,29,40,42)(28,44,46,34)(30,37,48,38);(2,8)(3,12)(4,5)(9,10)(11,15)(13,14)(17,18)(19,29)(20,36)(21,40)(22,26)(23,27)(24,33)(25,39)(28,45)(30,43)(31,42)(32,41)(34,37)(35,46)(38,44)(47,48)

Polycyclic presentation:

2^9|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=fhi,g^a=gi,c^b=cf,d^b=dg,e^b=ei,f^b=fh,g^b=gi,d^c=dh,e^c=ehi,f^c=fh

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 50725);