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 |
| prime radical | C22×D4 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
3906 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 48 points:
(2,6)(7,13)(8,11)(14,15)(17,18)(19,22)(20,30)(21,29)(23,28)(24,27)(25,32)(26,31)(33,34,38,41)(35,39,42,46)(36,37,40,45)(43,44,47,48);(1,2)(3,7)(4,8)(5,6)(9,14)(10,13)(11,12)(15,16)(17,19,23,27)(18,24,28,22)(20,29,32,26)(21,30,31,25)(35,42)(39,46)(43,47)(44,48);(17,20)(18,25)(19,29)(21,24)(22,31)(23,32)(26,27)(28,30)(33,35,38,42)(34,39,41,46)(36,43,40,47)(37,44,45,48);(1,2)(3,7)(4,11)(5,6)(8,12)(9,15)(10,13)(14,16)(17,21,23,31)(18,26,28,29)(19,25,27,30)(20,24,32,22)(33,36)(34,37)(35,43)(38,40)(39,44)(41,45)(42,47)(46,48);(1,3,5,10)(2,7,6,13)(4,9,12,16)(8,14,11,15)(17,22,23,24)(18,27,28,19)(20,31,32,21)(25,26,30,29);(1,4)(2,8)(3,9)(5,12)(6,11)(7,14)(10,16)(13,15)(17,18)(19,24)(20,25)(21,29)(22,27)(23,28)(26,31)(30,32)(33,37)(34,40)(35,44)(36,41)(38,45)(39,47)(42,48)(43,46)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=i,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,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, 8332596);