#6348492

order	512
nilpotency class	3
isoclinism	isoclinic class of #6348447
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	#62761	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2627 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)(3,10)(4,8)(5,13)(6,9)(7,11)(12,15)(14,16)(17,18,23,27)(19,28,31,38)(20,25,33,41)(21,26,22,35)(24,36,39,30)(29,43,46,48)(32,40,34,42)(37,44,47,45);(1,2)(3,7)(4,8)(5,13)(6,9)(10,14)(11,15)(12,16)(17,19,23,31)(18,24,27,39)(20,29,33,46)(21,30,22,36)(25,37,41,47)(26,38,35,28)(32,44,34,45)(40,43,42,48);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,16)(10,13)(17,20)(18,25)(19,29)(21,34)(22,32)(23,33)(24,37)(26,42)(27,41)(28,43)(30,45)(31,46)(35,40)(36,44)(38,48)(39,47);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(17,21,23,22)(18,26,27,35)(19,30,31,36)(20,32,33,34)(24,38,39,28)(25,40,41,42)(29,44,46,45)(37,43,47,48);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(18,27)(24,39)(25,41)(26,35)(28,38)(37,47)(40,42)(43,48)

Polycyclic presentation:

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

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, 6255688);