Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4308740 |
| automorphism | Aut(G) order 2097152 |
| 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 | |
1825 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(6,9)(10,12)(15,21)(16,23)(18,24)(19,26)(30,35)(32,37)(33,41)(38,44);(13,14,17,20)(15,18,21,24)(16,19,23,26)(22,25,27,28)(29,30,34,35)(31,37,39,32)(33,36,41,42)(38,43,44,40);(1,2)(3,4)(13,14)(15,18)(16,19)(17,20)(21,24)(22,25)(23,26)(27,28)(29,31,34,39)(30,32,35,37)(33,38,41,44)(36,40,42,43);(5,6)(7,10)(8,9)(11,12)(13,15,17,21)(14,18,20,24)(16,22,23,27)(19,25,26,28)(29,32)(30,31)(33,40)(34,37)(35,39)(36,38)(41,43)(42,44);(5,7,8,11)(6,10,9,12)(13,16)(14,19)(15,22)(17,23)(18,25)(20,26)(21,27)(24,28)(29,33,34,41)(30,36,35,42)(31,38,39,44)(32,40,37,43)Polycyclic presentation:
2^9|b^2=h,c^2=f,d^2=fh,e^2=fhi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^c=df80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4218923);