Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1979 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)(6,9)(10,12)(13,14)(15,21)(16,18)(17,24)(19,23)(20,26)(22,27)(25,28)(30,34)(31,37)(36,40)(39,43);(7,11)(10,12)(14,16)(19,21)(20,24)(27,28)(29,30,33,34)(31,38,37,35)(32,36,41,40)(39,44,43,42);(5,6)(7,10)(8,9)(11,12)(13,15,18,23)(14,19,16,21)(17,25,26,22)(20,28,24,27)(29,30)(31,35)(32,40)(33,34)(36,41)(37,38)(39,44)(42,43);(1,2)(3,4)(5,7)(6,10)(8,11)(9,12)(13,16,18,14)(15,21,23,19)(17,24,26,20)(22,28,25,27)(29,31,33,37)(30,35,34,38)(32,39,41,43)(36,42,40,44);(13,17,18,26)(14,20,16,24)(15,22,23,25)(19,27,21,28)(29,32,33,41)(30,36,34,40)(31,39,37,43)(35,42,38,44)Polycyclic presentation:
2^9|a^2=fi,b^2=h,c^2=fh,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4515225);