#134857

order	512
nilpotency class	4
isoclinism	isoclinic class of #134769
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1808
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#427
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 223 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) = 80
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 80 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dg,c^2=h,d^2=gi,e^2=i,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,e^c=ei,f^c=fhi

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, 42053);