#496439

order	512
nilpotency class	4
isoclinism	isoclinic class of #33764
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂²×#102
prime radical	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

700 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 80
transitive degree	μ_t(G) = 128

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403635);