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 | |
1806 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)(7,10)(8,9)(11,12)(13,14)(15,19)(16,18)(17,20)(21,23)(22,27)(24,26)(25,28)(29,30,34,35)(31,36,40,44)(32,41,37,33)(38,43,42,39);(5,7,9,12)(6,10,8,11)(14,16)(15,21)(20,24)(22,25)(30,35)(32,37)(36,44)(38,42);(6,8)(10,11)(13,15,18,21)(14,19,16,23)(17,25,26,22)(20,28,24,27)(29,31,34,40)(30,36,35,44)(32,42,37,38)(33,43,41,39);(1,2,3,4)(5,8,9,6)(7,11,12,10)(13,16,18,14)(15,19,21,23)(17,24,26,20)(22,27,25,28)(29,32,34,37)(30,33,35,41)(31,38,40,42)(36,39,44,43);(5,7)(6,10)(8,11)(9,12)(13,17,18,26)(14,20,16,24)(15,22,21,25)(19,27,23,28)(29,33)(30,37)(31,39)(32,35)(34,41)(36,42)(38,44)(40,43)Polycyclic presentation:
2^9|a^2=fh,b^2=g,c^2=f,d^2=i,e^2=h,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, 5189536);