Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #6131635 |
| automorphism | Aut(G) order 2097152 |
| 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 | |
1717 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)(7,11)(10,12)(15,21)(16,23)(18,24)(19,26)(29,30)(31,34)(32,37)(33,36)(35,40)(38,41)(39,43)(42,44);(1,2)(3,4)(6,9)(10,12)(15,21)(18,24)(22,27)(25,28)(30,31)(33,41)(35,37)(39,44);(5,6)(7,10)(8,9)(11,12)(13,14,17,20)(15,18,21,24)(16,19,23,26)(22,25,27,28)(29,31,34,30)(32,37,40,35)(33,38,41,36)(39,43,44,42);(13,15)(14,18)(16,22)(17,21)(19,25)(20,24)(23,27)(26,28)(29,32,34,40)(30,35,31,37)(33,39,41,44)(36,42,38,43);(5,7,8,11)(6,10,9,12)(13,16)(14,19)(15,22)(17,23)(18,25)(20,26)(21,27)(24,28)(29,33,34,41)(30,36,31,38)(32,39,40,44)(35,42,37,43)Polycyclic presentation:
2^9|a^2=fg,c^2=i,d^2=g,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dgi,e^b=eg68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6039792);