#134858

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

Polycyclic presentation:

2^9|a^2=f,b^2=dgi,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, 42054);