Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5035983 |
| 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 | |
1810 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)(13,14,18,19)(15,21,23,17)(16,24,25,20)(22,26,28,27)(29,30,33,36)(31,37,39,34)(32,40,41,35)(38,42,44,43);(8,12)(10,11)(14,19)(16,25)(17,21)(22,28)(29,30,33,36)(31,34,39,37)(32,40,41,35)(38,43,44,42);(6,7)(10,11)(13,15,18,23)(14,17,19,21)(16,22,25,28)(20,26,24,27)(29,30)(31,37)(32,35)(33,36)(34,39)(38,43)(40,41)(42,44);(1,2,3,4)(5,7,9,6)(8,11,12,10)(13,16,18,25)(14,20,19,24)(15,22,23,28)(17,26,21,27)(29,31,33,39)(30,34,36,37)(32,38,41,44)(35,42,40,43);(5,8)(6,10)(7,11)(9,12)(13,17)(14,15)(16,26)(18,21)(19,23)(20,22)(24,28)(25,27)(29,32,33,41)(30,35,36,40)(31,38,39,44)(34,42,37,43)Polycyclic presentation:
2^9|a^2=fh,b^2=h,c^2=f,d^2=i,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4990792);