#6351921

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
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2652 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)(19,29)(20,25)(21,27)(22,36)(23,28)(24,31)(26,34)(30,45)(32,40)(33,42)(35,43)(37,41)(38,47)(39,46)(44,48);(1,3)(2,4)(5,12)(6,13)(7,14)(8,11)(9,16)(10,15)(17,19,23,32)(18,24,28,41)(20,30,34,48)(21,31,22,37)(25,38,26,39)(27,40,36,29)(33,46,35,47)(42,45,43,44);(1,4)(2,3)(5,11)(6,16)(7,15)(8,12)(9,13)(10,14)(17,20)(18,25)(19,30)(21,33)(22,35)(23,34)(24,38)(26,28)(27,42)(29,44)(31,46)(32,48)(36,43)(37,47)(39,41)(40,45);(1,2)(3,4)(5,9)(6,8)(7,10)(11,13)(12,16)(14,15)(17,20)(18,26)(19,30)(21,35)(22,33)(23,34)(24,39)(25,28)(27,42)(29,44)(31,47)(32,48)(36,43)(37,46)(38,41)(40,45);(1,5,7,6)(2,8,10,9)(3,12,14,13)(4,11,15,16)(17,21,23,22)(18,27,28,36)(19,31,32,37)(20,33,34,35)(24,40,41,29)(25,42,26,43)(30,46,48,47)(38,45,39,44)

Polycyclic presentation:

2^9|b^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, 6259117);