#6350111

order	512
nilpotency class	3
isoclinism	isoclinic class of #6350061
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2763 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|b^2=i,c^2=h,d^2=h,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fh,g^a=gi,f^b=fh,g^b=gi,f^c=fi,g^c=gi,e^d=eh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6257307);