#6357551

order	512
nilpotency class	3
isoclinism	isoclinic class of #6357516
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#127
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1842 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2,7,11)(3,9,13,16)(4,14,12,8)(5,10,6,15)(17,18,23,27)(19,28,31,38)(20,25,33,41)(21,26,22,36)(24,35,39,30)(29,42,44,47)(32,40,34,46)(37,45,48,43);(1,3,6,12,7,13,5,4)(2,8,10,9,11,14,15,16)(17,19,22,30,23,31,21,35)(18,24,26,28,27,39,36,38)(20,29,34,43,33,44,32,45)(25,37,40,42,41,48,46,47);(1,4,5,13,7,12,6,3)(2,9,15,8,11,16,10,14)(17,20,23,33)(18,25,27,41)(19,29,31,44)(21,32,22,34)(24,37,39,48)(26,40,36,46)(28,42,38,47)(30,43,35,45);(1,5,7,6)(2,10,11,15)(3,4,13,12)(8,9,14,16)(17,21,23,22)(18,26,27,36)(19,30,31,35)(20,32,33,34)(24,38,39,28)(25,40,41,46)(29,43,44,45)(37,47,48,42);(1,5,7,6)(2,10,11,15)(3,12,13,4)(8,16,14,9)(19,31)(20,33)(24,39)(25,41)(28,38)(30,35)(32,34)(40,46)

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=gh,d^2=h,e^2=i,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,d^b=dhi,e^b=eh,e^c=eh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6264747);