#6351989

order	512
nilpotency class	3
isoclinism	isoclinic class of #6351885
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	#63056	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2612 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)(3,11)(4,13)(5,8)(6,9)(7,10)(12,15)(14,16)(17,18,23,27)(19,28,31,39)(20,32,34,25)(21,26,22,36)(24,37,40,30)(29,44,47,43)(33,42,35,41)(38,46,48,45);(1,3)(2,4)(5,12)(6,13)(7,14)(8,11)(9,16)(10,15)(17,19,23,31)(18,24,27,40)(20,29,34,47)(21,30,22,37)(25,38,32,48)(26,39,36,28)(33,45,35,46)(41,43,42,44);(1,4)(2,3)(5,11)(6,16)(7,15)(8,12)(9,13)(10,14)(17,18)(19,24)(20,25)(21,26)(22,36)(23,27)(28,37)(29,38)(30,39)(31,40)(32,34)(33,41)(35,42)(43,45)(44,46)(47,48);(1,2)(3,4)(5,9)(6,8)(7,10)(11,13)(12,16)(14,15)(17,20,23,34)(18,25,27,32)(19,29,31,47)(21,35,22,33)(24,38,40,48)(26,42,36,41)(28,43,39,44)(30,46,37,45);(1,5,7,6)(2,8,10,9)(3,12,14,13)(4,11,15,16)(17,21,23,22)(18,26,27,36)(19,30,31,37)(20,33,34,35)(24,39,40,28)(25,41,32,42)(29,45,47,46)(38,43,48,44)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6259185);