#6352000

order	512
nilpotency class	3
isoclinism	isoclinic class of #6351885
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2626 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6259196);