#135641

order	512
nilpotency class	3
isoclinism	isoclinic class of #120908
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#382
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1832
prime radical	C₂³	#433
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

152 conjugacy classes of subgroups.

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

Polycyclic presentation:

2^9|a^2=f,b^2=e,c^2=d,d^2=i,e^2=gi,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,c^b=chi,d^b=di,f^b=fgi,e^c=ei,f^c=fgi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 42837);