#6365656

order	512
nilpotency class	3
isoclinism	isoclinic class of #6365522
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1530 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6272852);