#135692

order	512
nilpotency class	4
isoclinism	isoclinic class of #135691
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1835
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#488
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ 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)(3,11,15,7)(4,16,18,31)(5,10)(6,9)(8,17,22,26)(12,19)(13,27)(14,24,25,21)(20,28)(23,32)(29,30)(33,34)(35,48,52,64)(36,55)(37,60)(38,46)(39,45)(40,47)(41,51,65,83)(42,56)(43,61)(44,49)(50,78,80,91)(53,75,54,74)(57,87)(58,86)(59,68)(62,72)(63,81,90,82)(66,76,67,84)(69,77)(70,88)(71,89)(73,92,85,79)(93,96,94,95);(1,3,12,25,6,15,5,14)(2,7,19,24,9,11,10,21)(4,13,26,32,18,29,17,28)(8,20,31,27,22,23,16,30)(33,35,49,76,39,54,62,83,40,52,61,84,38,53,37,51)(34,41,60,74,46,67,43,48,47,65,72,75,45,66,44,64)(36,50,77,95,58,82,89,92,59,80,88,96,57,81,56,79)(42,63,87,93,70,78,68,73,71,90,86,94,69,91,55,85);(1,4)(2,8,9,22)(3,13)(5,17)(6,18)(7,20,11,23)(10,16,19,31)(12,26)(14,28)(15,29)(21,30,24,27)(25,32)(33,36,37,56,38,57,61,88,40,59,62,89,39,58,49,77)(34,42,43,68,45,69,60,87,47,71,44,55,46,70,72,86)(35,50,51,79,53,81,84,96,52,80,83,92,54,82,76,95)(41,63,48,73,66,91,74,93,65,90,64,85,67,78,75,94)

Polycyclic presentation:

2^9|a^2=f,b^2=egi,c^2=d,d^2=hi,e^2=g,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,h^a=hi,f^b=fghi,f^c=fgh

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, 42888);