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 | |
4260 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2)(3,5)(4,8)(6,7)(9,13)(10,12)(11,14)(15,16)(17,18)(19,22)(20,23)(21,26)(24,29)(25,28)(27,31)(30,32)(33,34,38,42)(35,43,44,39)(36,40,46,48)(37,41,45,47);(4,11)(8,14)(10,15)(12,16)(17,18,22,19)(20,25,28,23)(21,24,29,26)(27,31,32,30)(35,44)(37,45)(39,43)(41,47);(2,7)(3,9)(8,14)(10,15)(18,19)(23,25)(24,26)(30,31)(33,35,38,44)(34,39,42,43)(36,45,46,37)(40,41,48,47);(1,3,6,9)(2,5,7,13)(4,10,11,15)(8,12,14,16)(17,19,22,18)(20,25,28,23)(21,26,29,24)(27,31,32,30)(33,36,38,46)(34,40,42,48)(35,37,44,45)(39,47,43,41);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,20)(18,23)(19,25)(21,27)(22,28)(24,30)(26,31)(29,32)(33,37)(34,41)(35,36)(38,45)(39,48)(40,43)(42,47)(44,46);(1,5)(2,9)(3,7)(4,12)(6,13)(8,15)(10,14)(11,16)(17,21,22,29)(18,24,19,26)(20,27,28,32)(23,30,25,31)(33,34)(35,39)(36,40)(37,47)(38,42)(41,45)(43,44)(46,48)Polycyclic presentation:
2^9|a^2=gi,b^2=g,c^2=gi,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10013778);