Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4311730 |
| 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 | |
1840 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)(14,18)(15,21)(16,23)(26,28)(29,30,34,35)(31,37,38,32)(33,41,42,36)(39,40,44,43);(5,6,8,10)(7,9,11,12)(14,18)(19,24)(20,25)(26,28)(30,35)(31,38)(33,42)(40,43);(5,7)(6,9)(8,11)(10,12)(13,14)(15,19)(16,20)(17,18)(21,24)(22,26)(23,25)(27,28)(29,31,34,38)(30,32,35,37)(33,39,42,44)(36,40,41,43);(1,2)(3,4)(5,7)(6,9)(8,11)(10,12)(13,15,17,21)(14,19,18,24)(16,22,23,27)(20,26,25,28)(29,32)(30,31)(33,40)(34,37)(35,38)(36,39)(41,44)(42,43);(13,16)(14,20)(15,22)(17,23)(18,25)(19,26)(21,27)(24,28)(29,33,34,42)(30,36,35,41)(31,39,38,44)(32,40,37,43)Polycyclic presentation:
2^9|a^2=f,b^2=gh,c^2=f,d^2=fh,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^b=ef,d^c=df80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4221707);