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 | #45101 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1859 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6,8,10)(7,11,12,9)(13,14)(15,27)(16,31)(17,25)(18,21)(19,23)(20,26)(22,35)(24,30)(28,44)(29,39)(32,42)(33,41)(34,38)(36,43)(37,40);(1,2)(3,4)(7,12)(9,11)(13,15,17,29,20,30,19,18)(14,21,23,24,26,39,25,27)(16,32,36,37,35,28,33,34)(22,40,43,42,31,38,41,44);(5,7,8,12)(6,9,10,11)(13,16,20,35)(14,22,26,31)(15,28,30,32)(17,36,19,33)(18,37,29,34)(21,38,39,40)(23,43,25,41)(24,44,27,42);(2,4)(6,10)(9,11)(13,17,20,19)(14,23,26,25)(15,29,30,18)(16,33,35,36)(21,24,39,27)(22,41,31,43)(28,37,32,34)(38,42,40,44);(13,18,19,30,20,29,17,15)(14,24,25,21,26,27,23,39)(16,34,33,28,35,37,36,32)(22,42,41,40,31,44,43,38)Polycyclic presentation:
2^9|a^2=gi,b^2=fi,c^2=g,d^2=i,e^2=f,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,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, 7445108);