#496390

order	512
nilpotency class	3
isoclinism	isoclinic class of #3198
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂²×D₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂²×C₂²⋊C₈
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

702 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)(17,18,22,28,23,29,43,56,24,30,44,57,21,27,41,54)(19,25,34,48,35,49,66,77,32,46,62,73,33,47,64,75)(20,36,38,67,37,52,68,55,39,53,42,69,40,26,50,51)(31,58,61,78,60,71,79,76,59,74,65,80,63,45,70,72);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,19)(18,25)(20,31)(21,33)(22,34)(23,35)(24,32)(26,45)(27,47)(28,48)(29,49)(30,46)(36,58)(37,60)(38,61)(39,59)(40,63)(41,64)(42,65)(43,66)(44,62)(50,70)(51,72)(52,71)(53,74)(54,75)(55,76)(56,77)(57,73)(67,78)(68,79)(69,80);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(17,20)(18,26)(19,31)(21,40)(22,42)(23,37)(24,39)(25,45)(27,53)(28,55)(29,36)(30,52)(32,59)(33,63)(34,65)(35,60)(38,44)(41,68)(43,50)(46,71)(47,74)(48,76)(49,58)(51,57)(54,67)(56,69)(61,62)(64,79)(66,70)(72,73)(75,78)(77,80);(2,8)(6,9)(7,16)(14,15)(19,32)(25,46)(31,59)(33,35)(34,62)(45,71)(47,49)(48,73)(58,74)(60,63)(61,65)(64,66)(70,79)(72,76)(75,77)(78,80)

Polycyclic presentation:

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

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, 403586);