#6468073

order	512
nilpotency class	3
isoclinism	isoclinic class of #6447056
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#555
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂	C₂×#2563
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1675 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=fgh,c^2=gh,d^2=h,g^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=dh,h^b=hi,e^c=ei

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, 6375269);