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 | |
1813 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)(10,12)(13,14,18,19)(15,20,21,16)(17,22,26,27)(23,25,28,24)(29,30,34,32)(31,37,39,35)(33,36,42,40)(38,44,41,43);(6,9)(10,12)(14,19)(15,21)(22,27)(23,28)(29,31,34,39)(30,35,32,37)(33,41,42,38)(36,44,40,43);(5,6,8,9)(7,10,11,12)(13,15)(14,20)(16,19)(17,23)(18,21)(22,25)(24,27)(26,28)(29,31)(30,35)(32,37)(33,38)(34,39)(36,43)(40,44)(41,42);(1,2)(3,4)(13,16,18,20)(14,21,19,15)(17,25,26,24)(22,23,27,28)(29,32,34,30)(31,37,39,35)(33,36,42,40)(38,43,41,44);(5,7)(6,10)(8,11)(9,12)(13,17)(14,22)(15,23)(16,24)(18,26)(19,27)(20,25)(21,28)(29,33,34,42)(30,36,32,40)(31,38,39,41)(35,43,37,44)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=fh,d^2=f,e^2=g,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, 5517616);