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 | |
4227 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)(3,7)(4,11)(5,6)(8,12)(9,13)(10,16)(14,15)(17,18)(19,26)(20,25)(21,23)(22,24)(27,32)(28,30)(29,31)(33,34)(35,44)(36,37)(38,48)(39,41)(40,45)(42,47)(43,46);(3,9)(7,13)(10,15)(14,16)(17,19,22,21)(18,23,24,26)(20,28,29,27)(25,32,31,30)(33,35,39,45)(34,40,41,44)(36,46,47,38)(37,48,42,43);(2,6)(7,13)(8,11)(14,16)(18,24)(23,26)(25,31)(30,32)(34,41)(35,45)(36,47)(43,48);(1,2)(3,7)(4,8)(5,6)(9,13)(10,14)(11,12)(15,16)(17,18)(19,23)(20,25)(21,26)(22,24)(27,30)(28,32)(29,31)(33,36)(34,42)(35,46)(37,41)(38,45)(39,47)(40,43)(44,48);(1,3)(2,7)(4,10)(5,9)(6,13)(8,14)(11,16)(12,15)(17,20,22,29)(18,25,24,31)(19,27,21,28)(23,30,26,32)(33,37,39,42)(34,36,41,47)(35,43,45,48)(38,44,46,40);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,21,22,19)(18,26,24,23)(20,27,29,28)(25,30,31,32)(33,38,39,46)(34,43,41,48)(35,36,45,47)(37,44,42,40)Polycyclic presentation:
2^9|b^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9621817);