#142833

order	512
nilpotency class	4
isoclinism	isoclinic class of #142831
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2011
commutator, Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #1729 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 623 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) = 72
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 72 points:

(1,2)(3,5)(4,7)(6,8)(9,10,17,24)(11,25,28,56)(12,19,37,52)(13,21,39,53)(14,20,38,54)(15,23,16,22)(18,41,43,27)(26,58,60,47)(29,46,63,45)(30,57,31,44)(32,65,66,42)(33,67,40,49)(34,55,68,48)(35,51,36,50)(59,70,64,71)(61,69,62,72);(2,5)(3,8)(6,7)(9,11)(10,18)(12,32)(13,29)(14,27)(15,31)(16,30)(17,28)(19,47)(20,44)(21,25)(22,46)(23,45)(24,43)(26,34)(33,62)(35,64)(36,59)(37,66)(38,41)(39,63)(40,61)(42,49)(48,71)(50,72)(51,69)(52,58)(53,56)(54,57)(55,70)(60,68)(65,67);(1,2)(3,6)(4,7)(5,8)(9,12,17,37)(10,19,24,52)(11,26,28,60)(13,34,39,68)(14,40,38,33)(15,36,16,35)(18,42,43,65)(20,49,54,67)(21,55,53,48)(22,51,23,50)(25,47,56,58)(27,32,41,66)(29,64,63,59)(30,62,31,61)(44,72,57,69)(45,71,46,70)

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 50029);