#6350163

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
prime radical	#63039	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2760 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=i,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, 6257359);