#6357733

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

1937 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=gi,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, 6264929);