#6356892

order	512
nilpotency class	3
isoclinism	isoclinic class of #6356733
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

1536 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fhi,c^2=i,d^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,d^b=dh,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, 6264088);