#6357548

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

Polycyclic presentation:

2^9|b^2=f,c^2=g,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, 6264744);