Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| 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 | |
1685 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)(7,11)(9,12)(13,14)(15,19)(16,18)(17,24)(20,26)(21,28)(22,23)(25,27)(29,30,34,33)(31,37,38,35)(32,40,41,36)(39,42,44,43);(5,6,8,10)(7,9,11,12)(14,16)(19,23)(20,24)(27,28)(30,33)(31,38)(32,41)(42,43);(5,6)(7,9)(8,10)(11,12)(13,15,18,22)(14,19,16,23)(17,25,26,21)(20,28,24,27)(29,31,34,38)(30,35,33,37)(32,39,41,44)(36,42,40,43);(1,2,3,4)(13,16,18,14)(15,19,22,23)(17,24,26,20)(21,27,25,28)(29,32,34,41)(30,36,33,40)(31,39,38,44)(35,42,37,43);(5,7)(6,9)(8,11)(10,12)(13,17,18,26)(14,20,16,24)(15,21,22,25)(19,27,23,28)(29,33,34,30)(31,35,38,37)(32,40,41,36)(39,42,44,43)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=f,d^2=i,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,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, 5185487);