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 | |
4182 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(1,2,5,8)(3,9,11,6)(4,7,13,12)(10,15,16,14)(20,28)(24,31)(27,32)(29,30)(34,37)(35,43)(36,46)(42,48);(1,2)(3,6)(4,7)(5,8)(9,11)(10,14)(12,13)(15,16)(17,18)(19,23)(20,24)(21,25)(22,26)(27,30)(28,31)(29,32)(33,34,39,37)(35,44,43,40)(36,47,46,41)(38,42,45,48);(17,19)(18,21)(20,27)(22,25)(23,26)(24,29)(28,32)(30,31)(33,35)(34,40)(36,45)(37,44)(38,46)(39,43)(41,48)(42,47);(1,2)(3,6)(4,12)(5,8)(7,13)(9,11)(10,15)(14,16)(17,19)(18,23)(20,27)(21,26)(22,25)(24,30)(28,32)(29,31)(33,36,39,46)(34,41,37,47)(35,45,43,38)(40,48,44,42);(1,3,5,11)(2,6,8,9)(4,10,13,16)(7,14,12,15)(17,20,22,28)(18,24,26,31)(19,27,25,32)(21,29,23,30)(33,37,39,34)(35,44,43,40)(36,47,46,41)(38,48,45,42);(1,4)(2,7)(3,10)(5,13)(6,14)(8,12)(9,15)(11,16)(17,21,22,23)(18,25,26,19)(20,29,28,30)(24,32,31,27)(33,38,39,45)(34,42,37,48)(35,46,43,36)(40,47,44,41)Polycyclic presentation:
2^9|a^2=gh,b^2=h,d^2=h,e^2=ghi,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8366294);