Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3069037 |
| automorphism | Aut(G) order 1048576 |
| 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 | |
1671 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(7,11)(10,12)(13,14)(15,21)(16,24)(17,18)(19,23)(20,25)(22,27)(26,28)(29,30,34,32)(31,37,39,35)(33,36,41,40)(38,43,44,42);(1,2,3,4)(14,18)(16,25)(19,21)(22,26)(30,32)(33,41)(35,37)(38,44);(1,2)(3,4)(5,6,9,8)(7,10,11,12)(13,14)(15,21)(16,20)(17,18)(19,23)(22,28)(24,25)(26,27)(29,31,34,39)(30,35,32,37)(33,38,41,44)(36,42,40,43);(5,7)(6,10)(8,12)(9,11)(13,15,17,23)(14,19,18,21)(16,26,25,22)(20,28,24,27)(29,32,34,30)(31,35,39,37)(33,36,41,40)(38,43,44,42);(5,8,9,6)(7,12,11,10)(13,16,17,25)(14,20,18,24)(15,22,23,26)(19,27,21,28)(29,33,34,41)(30,36,32,40)(31,38,39,44)(35,42,37,43)Polycyclic presentation:
2^9|a^2=g,b^2=fgh,c^2=fi,d^2=f,e^2=gi,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3144279);