#6356690

order	512
nilpotency class	3
isoclinism	isoclinic class of #6356633
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
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1528 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6263886);