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 | |
4115 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)(3,6)(4,8)(5,9)(7,10)(11,14)(12,13)(15,16)(17,18,22,23)(19,25,26,21)(20,28,30,24)(27,31,32,29)(33,34)(35,41)(36,39)(37,40)(38,43)(42,48)(44,46)(45,47);(1,3)(2,6)(4,11)(5,10)(7,9)(8,14)(12,15)(13,16)(17,18,22,23)(19,25,26,21)(20,24,30,28)(27,29,32,31)(33,34,37,40)(35,38,43,41)(36,39,46,44)(42,47,45,48);(3,10)(6,7)(11,16)(14,15)(18,23)(19,26)(24,28)(27,32)(33,34,37,40)(35,38,43,41)(36,39,46,44)(42,47,45,48);(1,3)(2,7)(4,11)(5,10)(6,9)(8,15)(12,14)(13,16)(17,19,22,26)(18,21,23,25)(20,27,30,32)(24,31,28,29)(33,34)(35,38)(36,44)(37,40)(39,46)(41,43)(42,48)(45,47);(1,4)(2,8)(3,11)(5,13)(6,14)(7,15)(9,12)(10,16)(17,20,22,30)(18,24,23,28)(19,27,26,32)(21,31,25,29)(33,35)(34,38)(36,45)(37,43)(39,48)(40,41)(42,46)(44,47);(1,2)(3,6)(4,12)(5,9)(7,10)(8,13)(11,15)(14,16)(17,21)(18,19)(20,29)(22,25)(23,26)(24,32)(27,28)(30,31)(33,36,37,46)(34,39,40,44)(35,42,43,45)(38,47,41,48)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=g,d^2=gh,e^2=gh,f^2=g,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, 8635614);