#513131

order	512
nilpotency class	5
isoclinism	isoclinic class of #513086
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂⁴
prime radical	C₂³	C₂×D₁₆
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

297 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 297 conjugacy classes of subgroups — too many to draw. Only its 118 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)(17,18,23,29)(19,30,35,61)(20,25,38,28)(21,39,42,73)(22,43,45,51)(24,46,49,32)(26,52,55,41)(27,56,58,37)(31,60,34,67)(33,65,66,79)(36,68,71,72)(40,57,75,80)(44,53,63,78)(47,64,59,62)(48,69,76,74)(50,77,54,70);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(17,19)(18,24)(20,31)(21,40)(22,44)(23,35)(25,47)(26,53)(27,57)(28,59)(29,49)(30,39)(32,41)(33,37)(34,38)(36,69)(42,75)(43,48)(45,63)(46,52)(50,65)(51,76)(54,79)(55,78)(56,66)(58,80)(60,68)(61,73)(62,70)(64,77)(67,72)(71,74);(2,8)(6,9)(7,16)(14,15)(17,20,23,38)(18,25,29,28)(19,31,35,34)(21,36,42,71)(22,37,45,56)(24,47,49,59)(26,50,55,54)(27,51,58,43)(30,60,61,67)(32,62,46,64)(33,63,66,44)(39,68,73,72)(40,69,75,74)(41,70,52,77)(48,57,76,80)(53,65,78,79)

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=e,d^2=i,f^2=gh,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fg,g^a=gh,h^a=hi,c^b=ce,f^c=fg,g^c=gh,h^c=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420327);