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 | |
1681 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)(6,9)(10,11)(13,14,18,19)(15,20,24,27)(16,17,25,21)(22,26,28,23)(29,30,35,38)(31,36,32,33)(34,37,42,44)(39,43,40,41);(5,6)(7,11)(8,9)(10,12)(14,19)(17,21)(20,27)(23,26)(29,31,35,32)(30,36,38,33)(34,40,42,39)(37,41,44,43);(5,7,8,12)(6,10,9,11)(13,15,18,24)(14,20,19,27)(16,22,25,28)(17,26,21,23)(29,32,35,31)(30,36,38,33)(34,40,42,39)(37,43,44,41);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,16)(14,21)(15,22)(17,19)(18,25)(20,23)(24,28)(26,27)(29,33,35,36)(30,32,38,31)(34,41,42,43)(37,40,44,39);(5,7)(6,10)(8,12)(9,11)(13,17,18,21)(14,16,19,25)(15,23,24,26)(20,28,27,22)(29,34)(30,37)(31,39)(32,40)(33,41)(35,42)(36,43)(38,44)Polycyclic presentation:
2^9|a^2=fgh,b^2=g,c^2=fg,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, 5417136);