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 | |
3939 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)(3,9)(4,13)(5,14)(6,8)(7,12)(10,11)(15,16)(17,18,23,28)(19,24,30,21)(20,26,32,25)(22,31,29,27)(34,40)(36,44)(41,46)(43,45);(1,3)(2,7)(4,10)(5,11)(6,12)(8,9)(13,15)(14,16)(17,19,23,30)(18,24,28,21)(20,29,32,22)(25,31,26,27)(33,34,39,40)(35,41,37,46)(36,42,44,47)(38,43,48,45);(2,8)(4,14)(7,9)(10,16)(17,20,23,32)(18,25,28,26)(19,22,30,29)(21,31,24,27)(33,35,39,37)(34,41,40,46)(36,43,44,45)(38,42,48,47);(1,4)(2,5)(3,10)(6,14)(7,11)(8,13)(9,15)(12,16)(17,20)(18,26)(19,29)(21,31)(22,30)(23,32)(24,27)(25,28)(33,36,39,44)(34,42,40,47)(35,43,37,45)(38,41,48,46);(1,3)(2,9)(4,10)(5,15)(6,12)(7,8)(11,13)(14,16)(17,21)(18,19)(20,31)(22,25)(23,24)(26,29)(27,32)(28,30)(33,37,39,35)(34,41,40,46)(36,45,44,43)(38,42,48,47);(1,5)(2,4)(3,11)(6,13)(7,10)(8,14)(9,16)(12,15)(17,22,23,29)(18,27,28,31)(19,20,30,32)(21,26,24,25)(33,38,39,48)(34,43,40,45)(35,42,37,47)(36,41,44,46)Polycyclic presentation:
2^9|a^2=i,b^2=gh,c^2=gh,d^2=ghi,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,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, 9770607);