Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519008 |
| 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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1682 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6,9,12)(7,11,8,10)(13,14)(15,24)(16,27)(17,31)(18,22)(19,34)(20,26)(21,30)(23,33)(25,40)(28,39)(29,43)(32,38)(35,44)(36,42)(37,41);(1,2)(3,4)(5,7,9,8)(6,10,12,11)(13,15,18,26)(14,20,22,24)(16,28,29,44)(17,19,32,33)(21,36,37,40)(23,38,34,31)(25,41,42,30)(27,35,43,39);(13,16,19,30,18,29,33,41)(14,21,23,27,22,37,34,43)(15,25,17,28,26,42,32,44)(20,35,31,36,24,39,38,40);(2,4)(13,17,19,26,18,32,33,15)(14,20,23,31,22,24,34,38)(16,28,30,42,29,44,41,25)(21,35,27,36,37,39,43,40);(5,8,9,7)(6,11,12,10)(14,22)(16,29)(20,24)(23,34)(25,42)(28,44)(30,41)(31,38)Polycyclic presentation:
2^9|a^2=fi,b^2=f,c^2=g,d^2=g,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,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7470795);