#146130

order	512
nilpotency class	4
isoclinism	isoclinic class of #146120
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2076
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#482
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

172 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 54 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) = 96
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 96 points:

(1,2)(3,11,14,24)(4,8)(5,9)(6,10)(7,13,21,18)(12,27,15,26)(16,22)(17,23)(19,25)(20,29,31,28)(30,32)(33,34,37,42,57,43,73,70,39,45,59,72,38,44,58,71)(35,40,47,62,48,74,80,95,50,64,82,92,49,63,81,91)(36,51,54,67,60,85,89,66,53,84,88,68,55,69,56,41)(46,75,77,93,52,86,87,90,76,94,96,65,78,83,79,61);(1,3,6,14)(2,7,10,21)(4,15,17,12)(5,18,19,13)(8,20,23,31)(9,24,25,11)(16,28,30,29)(22,27,32,26)(33,35)(34,40)(36,52)(37,47)(38,49)(39,50)(41,65)(42,62)(43,74)(44,63)(45,64)(46,55)(48,57)(51,83)(53,78)(54,87)(56,77)(58,81)(59,82)(60,76)(61,67)(66,93)(68,90)(69,94)(70,95)(71,91)(72,92)(73,80)(75,85)(79,88)(84,86)(89,96);(1,4,5,16,6,17,19,30)(2,8,9,22,10,23,25,32)(3,12,13,28,14,15,18,29)(7,20,11,26,21,31,24,27)(33,36,38,60,39,53,57,55)(34,41,44,67,45,66,43,68)(35,46,49,52,50,76,48,78)(37,56,58,54,59,89,73,88)(40,61,63,93,64,90,74,65)(42,69,71,51,72,85,70,84)(47,79,81,77,82,87,80,96)(62,83,91,75,92,86,95,94)

Polycyclic presentation:

2^9|a^2=e,b^2=g,c^2=dh,d^2=g,e^2=hi,f^2=i,h^2=i,b^a=bd,c^a=ce,f^a=fgi,c^b=cf,d^b=dg,e^b=eg,f^b=fi,d^c=di,e^c=eh,f^c=fgi,h^c=hi

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, 53326);