#6356806

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
prime radical	#3482	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1487 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)(3,10)(4,8)(5,15)(6,9)(7,13)(11,12)(14,16)(17,18,23,28)(19,29,31,40)(20,34,36,42)(21,27,22,26)(24,33,39,32)(25,37,41,35)(30,44,45,38)(43,47,48,46);(1,3,4,12,6,14,5,13)(2,7,15,16,9,11,8,10)(17,19,22,33,23,31,21,32)(18,24,27,40,28,39,26,29)(20,30,37,47,36,45,35,46)(25,38,42,48,41,44,34,43);(17,20,23,36)(18,25,28,41)(19,30,31,45)(21,35,22,37)(24,38,39,44)(26,34,27,42)(29,43,40,48)(32,46,33,47);(1,2)(3,11)(4,15)(5,8)(6,9)(7,14)(10,12)(13,16)(19,31)(24,39)(29,40)(30,45)(32,33)(38,44)(43,48)(46,47);(1,4,6,5)(2,8,9,15)(3,12,14,13)(7,10,11,16)

Polycyclic presentation:

2^9|a^2=i,b^2=fh,c^2=i,e^2=hi,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, 6264002);