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 | |
1772 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)(10,12)(13,14,18,19)(15,22,25,20)(16,17,26,21)(23,27,28,24)(29,30)(31,37)(32,35)(33,36)(34,39)(38,43)(40,42)(41,44);(7,11)(10,12)(14,19)(17,21)(20,22)(24,27)(29,30,33,36)(31,34,39,37)(32,35,41,44)(38,42,40,43);(5,6,8,9)(7,10,11,12)(13,15,18,25)(14,20,19,22)(16,23,26,28)(17,24,21,27)(29,30)(31,37)(32,35)(33,36)(34,39)(38,43)(40,42)(41,44);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,16,18,26)(14,21,19,17)(15,23,25,28)(20,27,22,24)(29,31)(30,34)(32,40)(33,39)(35,43)(36,37)(38,41)(42,44);(5,7)(6,10)(8,11)(9,12)(13,17,18,21)(14,16,19,26)(15,24,25,27)(20,23,22,28)(29,32,33,41)(30,35,36,44)(31,38,39,40)(34,42,37,43)Polycyclic presentation:
2^9|a^2=fh,b^2=h,c^2=g,d^2=fh,e^2=f,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, 5524348);