Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5605735 |
| 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 | |
1756 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)(5,6,10,8)(7,12,9,11)(13,14,17,20)(15,21,22,18)(16,24,25,19)(23,26,28,27)(29,30,34,35)(31,32,37,36)(33,38,41,43)(39,40,44,42);(5,6,10,8)(7,11,9,12)(15,22)(16,25)(18,21)(19,24)(30,35)(31,37)(33,41)(40,42);(5,7,10,9)(6,11,8,12)(13,15)(14,18)(16,23)(17,22)(19,26)(20,21)(24,27)(25,28)(29,31,34,37)(30,36,35,32)(33,39,41,44)(38,42,43,40);(1,2)(3,4)(5,8,10,6)(7,11,9,12)(13,14)(15,21)(16,19)(17,20)(18,22)(23,27)(24,25)(26,28)(29,32,34,36)(30,37,35,31)(33,42,41,40)(38,39,43,44);(5,9,10,7)(6,11,8,12)(13,16)(14,19)(15,23)(17,25)(18,26)(20,24)(21,27)(22,28)(29,33)(30,38)(31,39)(32,40)(34,41)(35,43)(36,42)(37,44)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=fgh,d^2=fgh,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5544655);