#6346694

order	512
nilpotency class	3
isoclinism	isoclinic class of #6346647
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	#62752	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

2747 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=h,d^2=i,e^2=hi,f^2=h,b^a=bf,c^a=cg,e^a=eh,f^a=fh,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, 6253890);