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 | |
4239 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)(3,5)(4,8)(6,7)(9,13)(10,12)(11,14)(15,16)(17,18)(19,22)(20,23)(21,24)(25,26)(27,32)(28,31)(29,30)(34,40)(35,38)(36,42)(45,47);(4,11)(8,14)(10,15)(12,16)(17,19)(18,22)(20,26)(21,29)(23,25)(24,30)(27,31)(28,32)(33,34)(35,43)(36,37)(38,41)(39,40)(42,48)(44,45)(46,47);(2,7)(3,9)(8,14)(10,15)(18,20)(19,25)(24,28)(27,29)(33,35,39,38)(34,41,40,43)(36,46,42,44)(37,45,48,47);(1,3,6,9)(2,5,7,13)(4,10,11,15)(8,12,14,16)(17,20,23,18)(19,26,25,22)(21,28,31,24)(27,30,29,32)(33,36,39,42)(34,37,40,48)(35,44,38,46)(41,47,43,45);(1,4,6,11)(2,8,7,14)(3,10,9,15)(5,12,13,16)(17,21,23,31)(18,24,20,28)(19,27,25,29)(22,32,26,30)(33,37)(34,42)(35,45)(36,40)(38,47)(39,48)(41,44)(43,46);(1,5)(2,9)(3,7)(4,12)(6,13)(8,15)(10,14)(11,16)(17,22,23,26)(18,25,20,19)(21,30,31,32)(24,27,28,29)(33,38,39,35)(34,41,40,43)(36,46,42,44)(37,47,48,45)Polycyclic presentation:
2^9|c^2=g,d^2=h,e^2=gh,f^2=i,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, 9624772);