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)(15,21,22,19)(16,24,25,20)(23,26,28,27)(30,31)(35,37)(36,38)(42,43);(5,7)(6,10)(8,11)(9,12)(14,18)(15,22)(16,25)(26,27)(29,30,34,31)(32,37,40,35)(33,38,41,36)(39,42,44,43);(6,11)(7,12)(13,14)(15,19)(16,24)(17,18)(20,25)(21,22)(23,27)(26,28)(29,31,34,30)(32,37,40,35)(33,36,41,38)(39,42,44,43);(1,2)(3,4)(5,7)(6,8)(9,12)(10,11)(13,15,17,22)(14,19,18,21)(16,23,25,28)(20,26,24,27)(29,32)(30,35)(31,37)(33,39)(34,40)(36,42)(38,43)(41,44);(5,8)(6,7)(9,10)(11,12)(13,16,17,25)(14,20,18,24)(15,23,22,28)(19,26,21,27)(29,33,34,41)(30,36,31,38)(32,39,40,44)(35,42,37,43)Polycyclic presentation:
2^9|a^2=fgi,b^2=gh,c^2=gh,d^2=fgh,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, 4951508);