#6349381

order	512
nilpotency class	3
isoclinism	isoclinic class of #6349370
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	#62823	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2604 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,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,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, 6256577);