#6357477

order	512
nilpotency class	3
isoclinism	isoclinic class of #6357459
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

1884 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)(3,12)(4,14)(5,10)(6,16)(7,11)(8,9)(13,15)(17,18)(19,28)(20,25)(21,26)(22,36)(23,27)(24,31)(29,41)(30,38)(32,42)(33,39)(34,40)(35,46)(37,45)(43,47)(44,48);(1,3,6,9,7,14,5,13)(2,8,10,12,11,15,16,4)(17,19,22,31,23,32,21,30)(18,24,26,28,27,38,36,42)(20,29,35,45,34,44,33,43)(25,37,39,41,40,47,46,48);(1,4,6,8,7,12,5,15)(2,9,10,14,11,13,16,3)(17,20)(18,25)(19,29)(21,33)(22,35)(23,34)(24,37)(26,39)(27,40)(28,41)(30,43)(31,45)(32,44)(36,46)(38,47)(42,48);(1,5,7,6)(2,10,11,16)(3,9,14,13)(4,15,12,8)(17,21,23,22)(18,26,27,36)(19,30,32,31)(20,33,34,35)(24,28,38,42)(25,39,40,46)(29,43,44,45)(37,41,47,48);(1,5,7,6)(2,10,11,16)(3,13,14,9)(4,8,12,15)(20,34)(25,40)(29,44)(33,35)(37,47)(39,46)(41,48)(43,45)

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=di,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, 6264673);