Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| 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 | |
1828 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)(5,6,9,10)(7,8,11,12)(13,14)(15,21)(16,20)(17,18)(19,22)(23,28)(24,26)(25,27)(29,30,34,32)(31,35,37,38)(33,40,41,36)(39,44,43,42);(5,7,9,11)(6,8,10,12)(14,18)(15,22)(16,24)(27,28)(31,37)(35,38)(39,43)(42,44);(6,10)(7,11)(13,15)(14,19)(16,25)(17,22)(18,21)(20,28)(23,24)(26,27)(29,31,34,37)(30,35,32,38)(33,39,41,43)(36,42,40,44);(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,19)(16,20)(17,18)(21,22)(23,27)(24,26)(25,28)(29,32,34,30)(31,38,37,35)(33,40,41,36)(39,44,43,42);(5,8)(6,7)(9,12)(10,11)(13,16,17,24)(14,20,18,26)(15,23,22,25)(19,27,21,28)(29,33,34,41)(30,36,32,40)(31,39,37,43)(35,42,38,44)Polycyclic presentation:
2^9|a^2=fh,b^2=fg,c^2=gh,d^2=gh,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5135006);