#513136

order	512
nilpotency class	5
isoclinism	isoclinic class of #513092
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

296 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 116 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)(19,24,30,46)(20,25,37,50)(21,38,40,70)(22,41,39,28)(26,51,53,34)(27,54,55,35)(31,47,57,42)(32,64,58,78)(33,66,59,45)(36,56,67,49)(43,60,74,62)(44,65,73,61)(48,69,52,68)(63,72,77,75)(71,80,76,79);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,19)(18,24)(20,31)(21,32)(22,33)(23,30)(25,47)(26,43)(27,44)(28,45)(29,46)(34,62)(35,61)(36,63)(37,57)(38,64)(39,59)(40,58)(41,66)(42,50)(48,76)(49,75)(51,60)(52,71)(53,74)(54,65)(55,73)(56,72)(67,77)(68,80)(69,79)(70,78);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(18,25)(19,30)(20,34)(21,35)(22,39)(24,42)(26,27)(28,56)(29,50)(31,60)(32,65)(36,69)(37,51)(38,48)(40,54)(41,49)(43,73)(44,74)(45,75)(46,47)(52,70)(53,55)(57,62)(58,61)(63,80)(64,71)(66,72)(67,68)(76,78)(77,79);(2,8)(6,9)(7,16)(14,15)

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=e,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=cei,f^c=fg,g^c=gh,h^c=hi

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, 420332);