Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5024635 |
| automorphism | Aut(G) order 4194304 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
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 | |
1860 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2)(3,4)(5,6,9,7)(8,11,12,10)(15,21)(16,23)(19,24)(20,25)(29,30,34,37)(31,32,39,35)(33,41,44,36)(38,43,42,40);(14,18)(19,24)(20,25)(26,28)(29,31,34,39)(30,35,37,32)(33,38,44,42)(36,43,41,40);(2,4)(5,7,9,6)(8,11,12,10)(13,14,17,18)(15,19,21,24)(16,20,23,25)(22,26,27,28)(29,31)(30,32)(33,42)(34,39)(35,37)(36,43)(38,44)(40,41);(1,2)(3,4)(5,8)(6,10)(7,11)(9,12)(13,15)(14,19)(16,22)(17,21)(18,24)(20,26)(23,27)(25,28)(29,32)(30,31)(33,43)(34,35)(36,42)(37,39)(38,41)(40,44);(13,16,17,23)(14,20,18,25)(15,22,21,27)(19,26,24,28)(29,33,34,44)(30,36,37,41)(31,38,39,42)(32,40,35,43)Polycyclic presentation:
2^9|a^2=fhi,b^2=f,c^2=fi,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^c=dg,e^c=ef,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4942393);