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 | |
1675 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2,3,4)(5,6)(7,10)(8,12)(9,11)(13,14,17,18)(15,19,21,25)(16,23,24,20)(22,28,27,26)(30,31)(32,39)(36,38)(40,44);(5,7)(6,10)(8,11)(9,12)(14,18)(15,21)(16,24)(26,28)(29,30,34,31)(32,37,39,35)(33,38,41,36)(40,42,44,43);(6,11)(7,12)(13,14)(15,19)(16,23)(17,18)(20,24)(21,25)(22,28)(26,27)(29,31,34,30)(32,37,39,35)(33,36,41,38)(40,42,44,43);(1,2)(3,4)(5,7)(6,8)(9,12)(10,11)(13,15)(14,19)(16,22)(17,21)(18,25)(20,26)(23,28)(24,27)(29,32,34,39)(30,35,31,37)(33,40,41,44)(36,42,38,43);(5,8)(6,7)(9,10)(11,12)(13,16,17,24)(14,20,18,23)(15,22,21,27)(19,26,25,28)(29,33,34,41)(30,36,31,38)(32,40,39,44)(35,42,37,43)Polycyclic presentation:
2^9|a^2=gi,b^2=gh,c^2=gh,d^2=gh,e^2=f,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, 4951499);