#6351874

order	512
nilpotency class	3
isoclinism	isoclinic class of #6351869
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
prime radical	#62835	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2757 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)(3,9)(4,12)(5,7)(6,11)(8,13)(10,16)(14,15)(17,18,23,28)(19,29,31,40)(20,34,37,41)(21,27,22,26)(24,32,39,33)(25,35,42,36)(30,38,44,48)(43,45,47,46);(4,13)(8,12)(9,16)(11,14)(17,19)(18,24)(20,30)(21,32)(22,33)(23,31)(25,38)(26,40)(27,29)(28,39)(34,47)(35,45)(36,46)(37,44)(41,43)(42,48);(1,3)(2,6)(4,14)(5,10)(7,15)(8,16)(9,12)(11,13)(17,20,23,37)(18,25,28,42)(19,30,31,44)(21,36,22,35)(24,38,39,48)(26,34,27,41)(29,43,40,47)(32,46,33,45);(2,7)(6,15)(8,12)(9,16)(19,31)(24,39)(29,40)(30,44)(32,33)(38,48)(43,47)(45,46);(17,21,23,22)(18,26,28,27)(19,32,31,33)(20,35,37,36)(24,40,39,29)(25,41,42,34)(30,45,44,46)(38,43,48,47)

Polycyclic presentation:

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

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, 6259070);