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 | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1791 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)(7,11)(8,12)(9,10)(13,14)(15,22)(16,28)(17,32)(18,23)(19,35)(20,26)(21,31)(24,33)(25,41)(27,37)(29,40)(30,44)(34,36)(38,43)(39,42);(1,2)(3,4)(6,10)(11,12)(13,15,18,26)(14,20,23,22)(16,29,30,34)(17,33,27,19)(21,38,39,41)(24,32,35,37)(25,42,43,31)(28,36,44,40);(5,7,9,8)(6,11,10,12)(13,16,19,31,18,30,33,42)(14,21,24,28,23,39,35,44)(15,25,27,29,26,43,17,34)(20,36,37,38,22,40,32,41);(2,4)(5,7,9,8)(6,11,10,12)(13,17,19,15,18,27,33,26)(14,22,24,32,23,20,35,37)(16,29,31,43,30,34,42,25)(21,36,28,38,39,40,44,41);(5,8,9,7)(6,11,10,12)(14,23)(16,30)(20,22)(24,35)(25,43)(29,34)(31,42)(32,37)Polycyclic presentation:
2^9|b^2=i,c^2=fgi,d^2=fgi,e^2=fi,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, 7472685);