#480069

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	C₂²×#41	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

748 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)(17,18,22,28,23,29,43,56,24,30,44,57,21,27,41,54)(19,31,35,59,34,47,63,55,33,53,42,69,40,25,45,48)(20,26,38,51,39,52,68,77,36,49,64,73,37,50,66,75)(32,58,61,78,60,72,79,76,62,74,67,80,65,46,70,71);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,19)(18,25)(20,32)(21,40)(22,42)(23,34)(24,33)(26,46)(27,53)(28,55)(29,31)(30,47)(35,44)(36,62)(37,65)(38,67)(39,60)(41,63)(43,45)(48,57)(49,72)(50,74)(51,76)(52,58)(54,59)(56,69)(61,64)(66,79)(68,70)(71,73)(75,78)(77,80);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,20)(18,26)(19,32)(21,37)(22,38)(23,39)(24,36)(25,46)(27,50)(28,51)(29,52)(30,49)(31,58)(33,62)(34,60)(35,61)(40,65)(41,66)(42,67)(43,68)(44,64)(45,70)(47,72)(48,71)(53,74)(54,75)(55,76)(56,77)(57,73)(59,78)(63,79)(69,80);(19,33)(20,36)(25,47)(26,49)(31,53)(34,40)(35,42)(37,39)(38,64)(45,63)(48,55)(50,52)(51,73)(59,69)(66,68)(75,77)

Polycyclic presentation:

2^9|a^2=f,c^2=g,e^2=i,f^2=h,h^2=i,b^a=be,e^a=eg,d^b=di,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, 387265);