Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4152 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,6,7)(3,11,12,8)(4,9,14,16)(5,10,15,13)(17,18,21,22)(19,23,27,25)(20,28,29,24)(26,31,32,30)(33,34)(35,38)(36,43)(37,39)(40,44)(41,47)(42,45)(46,48);(2,7)(3,12)(9,16)(10,13)(17,18)(19,25)(20,24)(21,22)(23,27)(26,31)(28,29)(30,32)(33,35,37,42)(34,38,39,45)(36,41,44,48)(40,46,43,47);(18,22)(23,25)(24,28)(30,31)(34,39)(38,45)(40,43)(46,47);(1,3,6,12)(2,8,7,11)(4,13,14,10)(5,9,15,16)(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,29)(31,32)(33,34)(35,38)(36,40)(37,39)(41,46)(42,45)(43,44)(47,48);(1,4,6,14)(2,9,7,16)(3,10,12,13)(5,11,15,8)(17,19,21,27)(18,23,22,25)(20,26,29,32)(24,30,28,31)(33,35)(34,38)(36,41)(37,42)(39,45)(40,46)(43,47)(44,48);(1,5)(2,10)(3,9)(4,8)(6,15)(7,13)(11,14)(12,16)(17,20,21,29)(18,24,22,28)(19,26,27,32)(23,30,25,31)(33,36,37,44)(34,40,39,43)(35,41,42,48)(38,46,45,47)Polycyclic presentation:
2^9|a^2=g,b^2=hi,d^2=gi,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^d=egi,f^e=fgi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10154636);