#6351822

order	512
nilpotency class	3
isoclinism	isoclinic class of #6351817
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

2758 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)(19,29,32,39)(20,34,37,41)(21,27,22,26)(24,31,40,33)(25,35,42,36)(30,38,45,48)(43,44,47,46);(4,13)(8,12)(9,16)(11,14)(17,19)(18,24)(20,30)(21,31)(22,33)(23,32)(25,38)(26,39)(27,29)(28,40)(34,47)(35,44)(36,46)(37,45)(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)(19,30,32,45)(21,36,22,35)(24,38,40,48)(26,34,27,41)(29,43,39,47)(31,46,33,44);(2,7)(6,15)(8,12)(9,16)(17,21,23,22)(18,26,28,27)(19,31,32,33)(20,35,37,36)(24,39,40,29)(25,41,42,34)(30,44,45,46)(38,43,48,47);(19,32)(24,40)(29,39)(30,45)(31,33)(38,48)(43,47)(44,46)

Polycyclic presentation:

2^9|a^2=h,c^2=h,d^2=h,f^2=h,g^2=hi,b^a=bf,c^a=cg,d^a=dhi,f^a=fh,g^a=ghi,e^b=eh,f^b=fh,g^b=gi,d^c=dh,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, 6259018);