Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1504 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(3,7)(6,8)(11,15)(13,16)(17,18)(19,22)(20,25)(21,24)(23,27)(26,29)(28,31)(30,32)(33,34,38,37)(35,41,43,39)(36,40,44,45)(42,47,48,46);(2,5)(6,8)(9,10,12,14)(11,13,15,16)(18,19)(20,27)(24,26)(28,32)(36,44)(40,45)(42,48)(46,47);(1,2,4,5)(3,6,7,8)(9,10)(11,13)(12,14)(15,16)(17,19,22,18)(20,25,27,23)(21,24,29,26)(28,30,32,31)(33,35,38,43)(34,39,37,41)(36,42,44,48)(40,46,45,47);(9,11)(10,13)(12,15)(14,16)(17,20)(18,23)(19,25)(21,28)(22,27)(24,30)(26,31)(29,32)(33,36,38,44)(34,40,37,45)(35,42,43,48)(39,46,41,47);(1,3)(2,6)(4,7)(5,8)(9,11)(10,13)(12,15)(14,16)(17,21,22,29)(18,24,19,26)(20,28,27,32)(23,30,25,31)(33,37,38,34)(35,41,43,39)(36,45,44,40)(42,47,48,46)Polycyclic presentation:
2^9|a^2=fg,b^2=fi,c^2=ghi,d^2=fg,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4557712);