#8946216

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4125 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)(3,9)(4,7)(5,13)(6,8)(10,14)(11,15)(12,16)(17,18,23,21)(19,24,26,20)(22,29,30,25)(27,28,32,31)(33,34,39,38)(35,41,36,40)(37,42,46,45)(43,48,44,47);(1,3,6,4)(2,7,8,9)(5,11,14,12)(10,15,13,16)(17,19,23,26)(18,24,21,20)(22,27,30,32)(25,31,29,28)(33,35,39,36)(34,40,38,41)(37,43,46,44)(42,47,45,48);(2,8)(5,14)(7,9)(11,12)(18,21)(19,26)(22,30)(28,31)(33,35,39,36)(34,41,38,40)(37,44,46,43)(42,47,45,48);(1,4,6,3)(2,9,8,7)(5,12,14,11)(10,16,13,15)(17,20)(18,19)(21,26)(22,28)(23,24)(25,27)(29,32)(30,31)(33,36,39,35)(34,41,38,40)(37,43,46,44)(42,47,45,48);(1,2)(3,9)(4,7)(5,10)(6,8)(11,16)(12,15)(13,14)(17,21,23,18)(19,24,26,20)(22,29,30,25)(27,31,32,28)(33,37,39,46)(34,42,38,45)(35,43,36,44)(40,47,41,48);(1,5)(2,10)(3,11)(4,12)(6,14)(7,15)(8,13)(9,16)(17,22,23,30)(18,25,21,29)(19,27,26,32)(20,28,24,31)(33,38,39,34)(35,41,36,40)(37,45,46,42)(43,48,44,47)

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=ghi,d^2=g,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8853412);