#480199

order	512
nilpotency class	3
isoclinism	isoclinic class of #17603
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂²×C₂²⋊C₈
prime radical	#3455	C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

707 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)(4,7)(5,8)(10,15,16,13)(12,14)(17,18,22,28,23,29,44,58,24,30,45,59,21,27,42,56)(19,31,34,61,33,49,64,57,35,55,43,71,41,25,46,48)(20,26,39,53,40,54,70,78,37,51,68,76,38,52,69,77)(32,60,63,79,62,50,74,75,36,65,67,80,66,47,72,73);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,19)(18,25)(20,36)(21,41)(22,43)(23,33)(24,35)(26,50)(27,55)(28,57)(29,31)(30,49)(32,37)(34,45)(38,62)(39,63)(40,66)(42,64)(44,46)(47,51)(48,59)(52,60)(53,73)(54,65)(56,61)(58,71)(67,68)(69,72)(70,74)(75,76)(77,80)(78,79);(17,20,23,40,24,37,21,38)(18,26,29,54,30,51,27,52)(19,32,33,62,35,36,41,66)(22,39,44,70,45,68,42,69)(25,47,31,60,49,50,55,65)(28,53,58,78,59,76,56,77)(34,63,64,74,43,67,46,72)(48,73,61,79,57,75,71,80);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(20,37)(26,51)(32,36)(38,40)(39,68)(47,50)(52,54)(53,76)(60,65)(62,66)(63,67)(69,70)(72,74)(73,75)(77,78)(79,80)

Polycyclic presentation:

2^9|a^2=f,c^2=h,d^2=g,e^2=i,f^2=h,h^2=i,b^a=be,e^a=eg,c^b=ci,e^b=ei,f^b=fgi,d^c=di

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 387395);