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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1574 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(5,6)(7,11)(8,9)(10,12)(13,14,18,15)(16,21,24,19)(17,20,25,22)(23,27,28,26)(29,30)(31,36)(32,40)(33,41)(34,35)(37,38)(39,43)(42,44);(5,7,10,8)(6,11,12,9)(14,15)(17,25)(19,21)(23,28)(30,35)(33,40)(36,43)(38,42);(6,12)(7,8)(13,15,18,14)(16,21,24,19)(17,20,25,22)(23,26,28,27)(29,31,34,39)(30,36,35,43)(32,37,41,44)(33,42,40,38);(1,2,3,4)(5,8,10,7)(6,11,12,9)(13,16)(14,19)(15,21)(17,23)(18,24)(20,26)(22,27)(25,28)(29,32)(30,33)(31,37)(34,41)(35,40)(36,42)(38,43)(39,44);(5,9,10,11)(6,7,12,8)(13,17,18,25)(14,20,15,22)(16,23,24,28)(19,26,21,27)(29,33)(30,32)(31,38)(34,40)(35,41)(36,44)(37,43)(39,42)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=f,d^2=fi,e^2=h,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, 5416991);