#6349120

order	512
nilpotency class	3
isoclinism	isoclinic class of #6349099
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	C₂×#3547	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2750 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,d^2=i,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,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

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6256316);