#9009237

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#44438	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4029 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=h,d^2=i,e^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8916433);