#6349734

order	512
nilpotency class	3
isoclinism	isoclinic class of #6349684
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
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2703 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6256930);