#6357704

order	512
nilpotency class	3
isoclinism	isoclinic class of #6357668
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

1834 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)(3,5)(4,7)(6,10)(8,11)(9,12)(13,15)(14,16)(17,18)(19,28)(20,25)(21,34)(22,27)(23,30)(24,32)(26,35)(29,42)(31,43)(33,39)(36,48)(37,45)(38,46)(40,47)(41,44);(1,3,6,12,8,14,7,13)(2,9,4,5,11,15,10,16)(17,19,21,30,22,31,35,46)(18,23,26,28,27,38,34,43)(20,29,33,45,32,44,47,48)(24,36,39,41,25,37,40,42);(1,4,6,11,8,10,7,2)(3,5,12,15,14,16,13,9)(17,20)(18,24)(19,29)(21,33)(22,32)(23,36)(25,27)(26,39)(28,41)(30,45)(31,44)(34,40)(35,47)(37,38)(42,43)(46,48);(1,5)(2,3)(4,12)(6,15)(7,9)(8,16)(10,13)(11,14)(17,20)(18,25)(19,29)(21,33)(22,32)(23,37)(24,27)(26,40)(28,42)(30,45)(31,44)(34,39)(35,47)(36,38)(41,43)(46,48);(1,6,8,7)(2,10,11,4)(3,12,14,13)(5,9,16,15)(20,32)(24,25)(29,44)(33,47)(36,37)(39,40)(41,42)(45,48)

Polycyclic presentation:

2^9|b^2=f,c^2=gh,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,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, 6264900);