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 | |
3965 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)(3,4,9,12)(5,8,15,16)(10,11,13,14)(17,18)(19,23)(20,28)(21,25)(22,26)(24,29)(27,30)(31,32)(33,34)(35,41)(36,39)(37,40)(38,43)(42,47)(44,46)(45,48);(2,7)(3,9)(8,16)(10,13)(17,19)(18,23)(20,25)(21,28)(22,27)(24,32)(26,30)(29,31)(33,34,37,40)(35,38,43,41)(36,39,44,46)(42,45,48,47);(2,7)(3,9)(8,16)(10,13)(19,27)(20,29)(23,30)(24,28)(33,34)(35,38)(36,39)(37,40)(41,43)(42,45)(44,46)(47,48);(1,3)(2,4)(5,13)(6,9)(7,12)(8,14)(10,15)(11,16)(17,20,22,29)(18,24,26,28)(19,25,27,31)(21,23,32,30)(33,35,37,43)(34,38,40,41)(36,42,44,48)(39,45,46,47);(1,4)(2,3)(5,14)(6,12)(7,9)(8,13)(10,16)(11,15)(17,18)(19,23)(20,24)(21,31)(22,26)(25,32)(27,30)(28,29)(33,36,37,44)(34,39,40,46)(35,42,43,48)(38,45,41,47);(1,5,6,15)(2,8,7,16)(3,10,9,13)(4,11,12,14)(17,21,22,32)(18,25,26,31)(19,28,27,24)(20,30,29,23)Polycyclic presentation:
2^9|a^2=h,b^2=gh,d^2=i,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8657709);