#142821

order	512
nilpotency class	4
isoclinism	isoclinic class of #142236
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#447
commutator, Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ #1729 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

617 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 617 conjugacy classes of subgroups — too many to draw. Only its 80 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) = 76
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 76 points:

(1,2)(3,4)(5,6)(7,9)(8,11)(10,12)(13,14,20,25)(15,26,31,62)(16,22,36,55)(17,37,18,42)(19,46,40,24)(21,29,51,28)(23,45,58,41)(27,61,65,76)(30,68,47,56)(32,70,63,73)(33,57,34,71)(35,60,48,54)(38,59,43,50)(39,53,44,72)(49,67,74,66)(52,69,75,64);(6,9)(7,12)(10,11)(13,15,20,31)(14,21,25,51)(16,32,36,63)(17,38,18,43)(19,47,40,30)(22,52,55,75)(23,56,58,68)(24,59,46,50)(26,42,62,37)(27,33,65,34)(28,45,29,41)(35,66,48,67)(39,64,44,69)(49,72,74,53)(54,76,60,61)(57,70,71,73);(2,4)(5,6)(7,10)(8,11)(9,12)(13,16,20,36)(14,22,25,55)(15,27,31,65)(17,39,18,44)(19,48,40,35)(21,49,51,74)(23,57,58,71)(24,60,46,54)(26,61,62,76)(28,66,29,67)(30,69,47,64)(32,38,63,43)(33,45,34,41)(37,53,42,72)(50,70,59,73)(52,68,75,56)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 50017);