Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5473759 |
| 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 | |
1892 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)(15,21,22,16)(17,20,26,27)(23,28,25,24)(31,35)(32,39)(38,43)(40,44);(6,9)(10,12)(14,19)(15,22)(20,27)(23,25)(29,30,34,37)(31,32,35,39)(33,41,42,36)(38,44,43,40);(5,6,8,9)(7,10,11,12)(13,15,18,22)(14,16,19,21)(17,25,26,23)(20,28,27,24)(29,31)(30,32)(33,38)(34,35)(36,40)(37,39)(41,44)(42,43);(1,2)(3,4)(13,16)(14,15)(17,24)(18,21)(19,22)(20,23)(25,27)(26,28)(29,32)(30,35)(31,37)(33,40)(34,39)(36,43)(38,41)(42,44);(5,7)(6,10)(8,11)(9,12)(13,17,18,26)(14,20,19,27)(15,23,22,25)(16,24,21,28)(29,33,34,42)(30,36,37,41)(31,38,35,43)(32,40,39,44)Polycyclic presentation:
2^9|a^2=f,b^2=fg,c^2=h,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=efg,d^c=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5382382);