#6347395

order	512
nilpotency class	3
isoclinism	isoclinic class of #6347363
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	C₂²×C₄	C₂⁵
prime radical	#3444	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

2648 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6254591);