#494430

order	512
nilpotency class	3
isoclinism	isoclinic class of #3068
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	C₂×#121
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#1869
prime radical	C₂×#575	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1263 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401626);