#6357020

order	512
nilpotency class	3
isoclinism	isoclinic class of #6356901
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₄²	C₂⁵
prime radical	#62129	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1494 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fi,c^2=i,d^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,c^b=ch,d^b=dh,g^c=gi,e^d=ehi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6264216);