#6349757

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

2887 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)(3,8)(4,10)(5,9)(6,11)(7,12)(13,16)(14,15)(17,18)(19,29)(20,34)(21,28)(22,27)(23,26)(24,32)(25,37)(30,44)(31,39)(33,40)(35,42)(36,41)(38,45)(43,47)(46,48);(1,3)(2,7)(4,12)(5,13)(6,14)(8,9)(10,15)(11,16)(17,19)(18,24)(20,30)(21,33)(22,32)(23,31)(25,38)(26,40)(27,39)(28,29)(34,48)(35,45)(36,47)(37,46)(41,44)(42,43);(1,3)(2,8)(4,13)(5,12)(6,14)(7,9)(10,16)(11,15)(17,20)(18,25)(19,30)(21,37)(22,36)(23,35)(24,38)(26,41)(27,34)(28,42)(29,43)(31,45)(32,47)(33,46)(39,48)(40,44);(1,4,6,5)(2,9,11,10)(3,12,14,13)(7,8,16,15)(20,35)(25,41)(30,45)(34,42)(36,37)(38,44)(43,48)(46,47);(1,4,6,5)(2,9,11,10)(3,13,14,12)(7,15,16,8)(18,26)(19,31)(25,41)(27,28)(30,45)(32,33)(34,42)(46,47)

Polycyclic presentation:

2^9|d^2=hi,e^2=hi,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, 6256953);