#6347242

order	512
nilpotency class	3
isoclinism	isoclinic class of #6347203
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	#62748	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

2815 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:

(1,2,5,8)(3,7,10,14)(4,6,12,9)(11,13,16,15)(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);(2,6)(4,12)(7,13)(8,9)(11,16)(14,15)(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);(1,3)(2,7)(4,11)(5,10)(6,13)(8,14)(9,15)(12,16)(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);(3,10)(7,14)(11,16)(13,15)(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);(2,8)(6,9)(7,14)(13,15)(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|a^2=h,d^2=i,e^2=i,f^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fh,f^b=fh,g^b=gi,d^c=dh,f^c=fi

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, 6254438);