Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5035983 |
| automorphism | Aut(G) order 2097152 |
| 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 | |
1797 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)(7,12)(8,9)(10,11)(13,14,17,18)(15,19,23,21)(16,20,25,26)(22,27,24,28)(30,31)(32,39)(36,38)(40,44);(5,7)(6,9)(8,10)(11,12)(14,18)(19,21)(20,26)(27,28)(29,30,34,31)(32,37,39,35)(33,38,41,36)(40,42,44,43);(7,8)(9,12)(13,15)(14,19)(16,24)(17,23)(18,21)(20,28)(22,25)(26,27)(29,31,34,30)(32,37,39,35)(33,36,41,38)(40,42,44,43);(1,2,3,4)(5,8,10,7)(6,9,11,12)(13,14)(15,21)(16,20)(17,18)(19,23)(22,28)(24,27)(25,26)(29,32)(30,35)(31,37)(33,40)(34,39)(36,42)(38,43)(41,44);(5,9)(6,7)(8,11)(10,12)(13,16,17,25)(14,20,18,26)(15,22,23,24)(19,27,21,28)(29,33,34,41)(30,36,31,38)(32,40,39,44)(35,42,37,43)Polycyclic presentation:
2^9|a^2=fgh,b^2=g,c^2=g,d^2=fi,e^2=fh,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, 4990206);