#6356925

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

1517 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:

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

Polycyclic presentation:

2^9|b^2=fh,d^2=i,e^2=i,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, 6264121);