#482356

order	512
nilpotency class	4
isoclinism	isoclinic class of #482080
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#102
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3050	C₂²
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

712 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)(3,9,11,6)(4,8)(5,7)(10,15,16,13)(12,14)(17,18,22,28,24,30,23,29)(19,31,34,54,36,59,35,58)(20,26,39,52,37,50,40,53)(21,41,43,27,45,55,44,56)(25,42,47,33,49,64,48,65)(32,57,62,75,60,77,63,78)(38,67,69,51,66,76,70,74)(46,68,72,61,71,80,73,79);(1,3)(2,6)(4,10)(5,11)(7,9)(8,13)(12,16)(14,15)(17,19,24,36)(18,25,30,49)(20,32,37,60)(21,42,45,64)(22,35,23,34)(26,46,50,71)(27,54,56,58)(28,48,29,47)(31,41,59,55)(33,43,65,44)(38,68,66,80)(39,63,40,62)(51,75,74,78)(52,73,53,72)(57,67,77,76)(61,69,79,70);(2,7)(6,9)(8,14)(13,15)(17,20)(18,26)(19,32)(21,38)(22,39)(23,40)(24,37)(25,46)(27,51)(28,52)(29,53)(30,50)(31,57)(33,61)(34,62)(35,63)(36,60)(41,67)(42,68)(43,69)(44,70)(45,66)(47,72)(48,73)(49,71)(54,75)(55,76)(56,74)(58,78)(59,77)(64,80)(65,79);(1,4,5,12)(2,8,7,14)(3,10,11,16)(6,13,9,15)(20,37)(26,50)(32,60)(38,66)(39,40)(46,71)(51,74)(52,53)(57,77)(61,79)(62,63)(67,76)(68,80)(69,70)(72,73)(75,78)

Polycyclic presentation:

2^9|a^2=f,b^2=i,d^2=gh,e^2=hi,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cgh,e^a=eg,g^a=gi,h^a=hi,e^b=eh,f^b=fghi,g^b=gi,h^b=hi,d^c=di

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 389552);