#6349279

order	512
nilpotency class	3
isoclinism	isoclinic class of #6349215
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	C₂⁴	#120
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2692 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=i,e^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=dhi,f^a=fh,g^a=gi,e^b=eh,f^b=fh,g^b=gi,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, 6256475);