Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1360 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,5)(6,8)(10,13)(11,15)(17,18)(19,25)(20,22)(21,24)(23,26)(27,32)(28,30)(29,31)(33,34,39,37)(35,40,36,43)(38,41,46,45)(42,47,44,48);(10,13)(14,16)(18,20)(19,26)(24,28)(27,29)(33,35,39,36)(34,40,37,43)(38,42,46,44)(41,47,45,48);(9,10)(11,14)(12,13)(15,16)(17,19)(18,23)(20,25)(21,29)(22,26)(24,32)(27,30)(28,31)(33,36,39,35)(34,40,37,43)(38,44,46,42)(41,47,45,48);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16)(17,20,22,18)(19,25,26,23)(21,28,30,24)(27,32,29,31)(33,37,39,34)(35,40,36,43)(38,45,46,41)(42,47,44,48);(1,3,4,7)(2,6,5,8)(9,11,12,15)(10,14,13,16)(17,21,22,30)(18,24,20,28)(19,27,26,29)(23,31,25,32)(33,38,39,46)(34,41,37,45)(35,42,36,44)(40,47,43,48)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,c^2=gh,d^2=fgh,e^2=ghi,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, 4670148);