#6458482

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#555
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂	C₂×#2495
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1133 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=g,c^2=fg,d^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^c=ei,e^d=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, 6365678);