#512811

order	512
nilpotency class	2
isoclinism	isoclinic class of #560
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₆ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂×C₁₆	C₂⁴
commutator, socle	C₂²	C₂³×C₁₆
prime radical	C₂³	C₂²×C₁₆
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

287 conjugacy classes of subgroups; only the 230 normal subgroups are shown.

normal subgroups only

This group has 287 conjugacy classes of subgroups — too many to draw. Only its 230 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 — 4 generators on 80 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=e,c^2=e,f^2=g,g^2=h,h^2=i,b^a=be,c^a=ci,d^a=de,c^b=cei,d^b=di

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420007);