#6347538

order	512
nilpotency class	3
isoclinism	isoclinic class of #6347491
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	#35210	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

2771 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)(3,11)(4,10)(5,9)(6,8)(7,13)(12,15)(14,16)(17,18)(19,24)(20,25)(21,35)(22,26)(23,27)(28,41)(29,45)(30,37)(31,38)(32,46)(33,39)(34,40)(36,44)(42,47)(43,48);(1,3,6,14)(2,7,8,15)(4,13,5,12)(9,11,10,16)(17,19)(18,24)(20,28)(21,29)(22,30)(23,31)(25,36)(26,37)(27,38)(32,42)(33,43)(34,44)(35,45)(39,47)(40,41)(46,48);(17,20)(18,25)(19,28)(21,32)(22,33)(23,34)(24,36)(26,39)(27,40)(29,42)(30,43)(31,44)(35,46)(37,47)(38,41)(45,48);(2,8)(7,15)(9,10)(11,16)(17,21,23,22)(18,26,27,35)(19,29,31,30)(20,32,34,33)(24,37,38,45)(25,39,40,46)(28,42,44,43)(36,47,41,48);(1,4,6,5)(2,9,8,10)(3,12,14,13)(7,16,15,11)(17,22,23,21)(18,26,27,35)(19,30,31,29)(20,33,34,32)(24,37,38,45)(25,39,40,46)(28,43,44,42)(36,47,41,48)

Polycyclic presentation:

2^9|b^2=h,d^2=i,e^2=hi,f^2=h,b^a=bf,c^a=cg,d^a=dhi,e^a=eh,f^a=fh,e^b=eh,f^b=fh,g^b=gi,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, 6254734);