#6352056

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

2682 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,d^2=h,e^2=hi,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, 6259252);