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 | |
3741 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)(3,6)(4,5)(7,8)(9,13)(10,16)(11,14)(12,15)(17,18)(19,26)(20,24)(21,25)(22,27)(23,30)(28,32)(29,31)(33,34,39,43)(35,42,38,40)(36,37,41,48)(44,45,46,47);(1,3,7,10)(2,6,8,16)(4,11,13,15)(5,14,9,12)(17,19)(18,23)(20,28)(21,29)(22,30)(24,31)(25,32)(26,27)(33,35,39,38)(34,40,43,42)(36,44,41,46)(37,47,48,45);(2,8)(3,10)(4,13)(12,14)(17,20,22,21)(18,24,27,25)(19,28,30,29)(23,31,26,32)(33,35,39,38)(34,40,43,42)(36,46,41,44)(37,45,48,47);(1,4)(2,9)(3,11)(5,8)(6,12)(7,13)(10,15)(14,16)(17,21,22,20)(18,25,27,24)(19,29,30,28)(23,32,26,31)(33,36,39,41)(34,37,43,48)(35,44,38,46)(40,47,42,45);(1,5,7,9)(2,4,8,13)(3,12,10,14)(6,15,16,11)(17,19)(18,26)(20,28)(21,29)(22,30)(23,27)(24,32)(25,31)(33,37)(34,41)(35,45)(36,43)(38,47)(39,48)(40,44)(42,46);(1,6)(2,10)(3,8)(4,12)(5,11)(7,16)(9,15)(13,14)(17,18)(19,23)(20,24)(21,25)(22,27)(26,30)(28,31)(29,32)(33,38,39,35)(34,42,43,40)(36,46,41,44)(37,45,48,47)Polycyclic presentation:
2^9|a^2=hi,b^2=h,c^2=g,d^2=g,e^2=i,f^2=hi,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, 9626346);