#6357758

order	512
nilpotency class	3
isoclinism	isoclinic class of #6357716
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 ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1836 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)(3,10)(4,13)(5,8)(6,12)(7,9)(11,15)(14,16)(17,18)(19,24)(20,32)(21,26)(22,36)(23,27)(25,34)(28,41)(29,38)(30,39)(31,45)(33,40)(35,46)(37,44)(42,48)(43,47);(1,3,10,2,7,12,6,9)(4,14,16,13,5,11,15,8)(17,19)(18,24)(20,28)(21,30)(22,31)(23,29)(25,37)(26,39)(27,38)(32,41)(33,43)(34,44)(35,42)(36,45)(40,47)(46,48);(1,4,7,5)(2,8,9,13)(3,11,12,14)(6,15,10,16)(17,20,22,34,23,35,21,33)(18,25,26,32,27,40,36,46)(19,28,31,44,29,42,30,43)(24,37,39,41,38,47,45,48);(1,5,7,4)(2,8,9,13)(3,11,12,14)(6,16,10,15)(19,29)(24,38)(28,42)(30,31)(37,47)(39,45)(41,48)(43,44);(4,5)(8,13)(11,14)(15,16)(17,21,23,22)(18,26,27,36)(19,30,29,31)(20,33,35,34)(24,39,38,45)(25,32,40,46)(28,43,42,44)(37,41,47,48)

Polycyclic presentation:

2^9|b^2=fh,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,c^b=ch,d^b=di,e^c=eh,e^d=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, 6264954);