Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4379698 |
| 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 | |
1788 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)(7,11)(9,12)(13,14,18,16)(15,19,21,22)(17,24,25,20)(23,28,27,26)(29,30,34,35)(31,36,41,43)(32,33,38,37)(39,42,44,40);(5,6,8,10)(7,9,11,12)(15,21)(19,22)(23,27)(26,28)(30,35)(32,38)(36,43)(39,44);(1,2,3,4)(13,15,18,21)(14,19,16,22)(17,23,25,27)(20,26,24,28)(29,31)(30,36)(32,39)(33,42)(34,41)(35,43)(37,40)(38,44);(5,7,8,11)(6,9,10,12)(13,16,18,14)(15,22,21,19)(17,24,25,20)(23,28,27,26)(29,32,34,38)(30,37,35,33)(31,39,41,44)(36,40,43,42);(13,17,18,25)(14,20,16,24)(15,23,21,27)(19,26,22,28)(29,33,34,37)(30,38,35,32)(31,40,41,42)(36,39,43,44)Polycyclic presentation:
2^9|a^2=fh,b^2=fi,c^2=gh,d^2=hi,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^c=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4302625);