#137895

order	512
nilpotency class	3
isoclinism	isoclinic class of #9369
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₄²	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1869
prime radical	C₂³	#408
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

276 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 86 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)(4,15,17,8)(5,18)(6,10)(7,13)(9,20)(12,25,28,32)(14,26)(16,23,19,24)(21,27,31,29)(22,30)(33,34,38,44,39,45,59,72,40,46,60,73,37,43,57,70)(35,41,49,63,48,62,79,90,50,64,81,92,55,68,85,94)(36,51,54,83,53,66,84,71,52,69,58,86,56,42,65,67)(47,74,77,95,78,61,87,89,75,82,80,96,76,88,93,91);(1,3,5,13,6,14,20,30)(2,7,9,11,10,22,18,26)(4,12,19,29,17,28,16,27)(8,21,24,25,15,31,23,32)(33,35,38,49,39,48,59,79,40,50,60,81,37,55,57,85)(34,41,44,63,45,62,72,90,46,64,73,92,43,68,70,94)(36,47,58,80,53,78,65,93,52,75,54,77,56,76,84,87)(42,61,71,91,51,82,86,95,66,88,67,89,69,74,83,96);(1,4,6,17)(2,8,10,15)(3,12,14,28)(5,19,20,16)(7,21,22,31)(9,24,18,23)(11,25,26,32)(13,29,30,27)(33,36)(34,42)(35,47)(37,56)(38,58)(39,53)(40,52)(41,61)(43,69)(44,71)(45,51)(46,66)(48,78)(49,80)(50,75)(54,60)(55,76)(57,84)(59,65)(62,82)(63,91)(64,88)(67,73)(68,74)(70,83)(72,86)(77,81)(79,93)(85,87)(89,92)(90,95)(94,96)

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=g,d^2=g,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,e^b=egi,d^c=dgi,e^c=ei,f^c=fgi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 45091);