#134859

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

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,c^b=ci,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, 42055);