#496470

order	512
nilpotency class	3
isoclinism	isoclinic class of #23995
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#535
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#21851
prime radical	#3455	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

741 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

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

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, 403666);