#9016449

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	#40982	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4073 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)(17,18)(19,26)(20,28)(21,25)(22,23)(24,29)(27,31)(30,32)(33,34,38,43)(35,39,46,40)(36,41,42,37)(44,45,47,48);(1,2)(3,7)(4,8)(5,9)(6,10)(11,14)(12,15)(13,16)(17,19,23,25)(18,21,22,26)(20,27,29,32)(24,31,28,30)(33,35)(34,39)(36,47)(37,45)(38,46)(40,43)(41,48)(42,44);(3,8)(4,7)(5,13)(9,16)(18,22)(20,29)(21,26)(27,32)(33,35,38,46)(34,40,43,39)(36,47,42,44)(37,48,41,45);(1,3)(2,7)(4,10)(5,11)(6,8)(9,14)(12,16)(13,15)(17,20,23,29)(18,24,22,28)(19,27,25,32)(21,31,26,30)(33,35)(34,39)(36,47)(37,45)(38,46)(40,43)(41,48)(42,44);(1,4,6,7)(2,8,10,3)(5,12,13,14)(9,15,16,11)(17,21,23,26)(18,25,22,19)(20,30,29,31)(24,27,28,32)(33,36,38,42)(34,41,43,37)(35,44,46,47)(39,45,40,48);(1,5)(2,9)(3,11)(4,12)(6,13)(7,14)(8,15)(10,16)(17,22,23,18)(19,26,25,21)(20,28,29,24)(27,30,32,31)(33,37,38,41)(34,42,43,36)(35,45,46,48)(39,44,40,47)

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=i,d^2=hi,e^2=gi,f^2=h,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, 8923645);