#6352026

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

2623 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)(3,12,9,16)(4,14,13,15)(6,10,7,11)(17,18,23,20)(19,27,31,40)(21,32,34,25)(22,33,37,26)(24,36,28,30)(29,43,44,41)(35,39,47,46)(38,45,42,48);(1,3)(2,4)(5,13)(6,14)(7,15)(8,9)(10,12)(11,16)(17,19)(18,24)(20,28)(21,29)(22,36)(23,31)(25,38)(26,40)(27,33)(30,37)(32,42)(34,44)(35,48)(39,43)(41,46)(45,47);(1,4,8,13)(2,9,5,3)(6,12,7,16)(10,15,11,14)(17,20,23,18)(19,28,31,24)(21,32,34,25)(22,26,37,33)(27,36,40,30)(29,42,44,38)(35,39,47,46)(41,48,43,45);(1,5,8,2)(3,13,9,4)(6,10,7,11)(12,15,16,14)(17,21,23,34)(18,25,20,32)(19,29,31,44)(22,35,37,47)(24,38,28,42)(26,39,33,46)(27,41,40,43)(30,45,36,48);(1,6,8,7)(2,10,5,11)(3,14,9,15)(4,12,13,16)(21,34)(25,32)(29,44)(35,47)(38,42)(39,46)(41,43)(45,48)

Polycyclic presentation:

2^9|a^2=h,c^2=h,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, 6259222);