Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518964 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #2397 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2397 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2063 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,3)(5,6)(7,11)(8,9)(10,12)(13,14,20,25)(15,21,28,38)(16,29,31,42)(17,34,27,37)(18,24,19,23)(22,33,41,32)(26,39,43,44)(30,36,35,40);(1,2,4,3)(5,6)(7,10)(8,9)(11,12)(13,15,20,28)(14,21,25,38)(16,30,31,35)(17,19,27,18)(22,39,41,44)(23,37,24,34)(26,33,43,32)(29,36,42,40);(6,9)(10,11)(13,16,18,32,20,31,19,33)(14,22,23,29,25,41,24,42)(15,26,27,30,28,43,17,35)(21,36,34,39,38,40,37,44);(5,7,8,12)(6,10,9,11)(13,17,19,28,20,27,18,15)(14,21,24,37,25,38,23,34)(16,30,33,26,31,35,32,43)(22,40,42,39,41,36,29,44);(1,3,4,2)(16,31)(22,41)(26,43)(29,42)(30,35)(32,33)(36,40)(39,44)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=hi,d^2=gi,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,d^c=di,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7503984);