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 | |
1724 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)(7,11)(8,9)(10,12)(14,18)(15,21)(16,24)(26,28)(29,30,33,34)(31,35,37,41)(32,39,40,36)(38,44,43,42);(5,7,9,12)(6,10,8,11)(14,18)(16,24)(19,22)(23,27)(30,34)(31,37)(32,40)(42,44);(5,6,9,8)(7,10,12,11)(13,14,17,18)(15,22,21,19)(16,20,24,25)(23,28,27,26)(29,30)(31,35)(32,39)(33,34)(36,40)(37,41)(38,44)(42,43);(1,2)(3,4)(5,8,9,6)(7,10,12,11)(13,15)(14,19)(16,23)(17,21)(18,22)(20,26)(24,27)(25,28)(29,31,33,37)(30,35,34,41)(32,38,40,43)(36,42,39,44);(5,7)(6,11)(8,10)(9,12)(13,16)(14,20)(15,23)(17,24)(18,25)(19,26)(21,27)(22,28)(29,32)(30,36)(31,38)(33,40)(34,39)(35,42)(37,43)(41,44)Polycyclic presentation:
2^9|a^2=gi,b^2=gh,c^2=fh,d^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, 4949899);