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 | |
1789 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)(7,9)(8,10)(11,12)(13,14,18,21)(15,17,22,19)(16,24,27,20)(23,28,25,26)(30,31)(32,39)(33,41)(42,43);(8,12)(10,11)(15,22)(17,19)(23,25)(26,28)(29,30,34,31)(32,35,39,37)(33,36,41,38)(40,42,44,43);(6,7)(10,11)(13,15)(14,19)(16,25)(17,21)(18,22)(20,26)(23,27)(24,28)(29,31,34,30)(32,35,39,37)(33,36,41,38)(40,43,44,42);(1,2,3,4)(5,7,9,6)(8,11,12,10)(13,16,18,27)(14,20,21,24)(15,23,22,25)(17,26,19,28)(29,32)(30,35)(31,37)(33,40)(34,39)(36,42)(38,43)(41,44);(5,8)(6,10)(7,11)(9,12)(13,17,18,19)(14,15,21,22)(16,26,27,28)(20,23,24,25)(29,33,34,41)(30,36,31,38)(32,40,39,44)(35,42,37,43)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=f,d^2=fi,e^2=fh,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, 5416913);