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 | C2×#469 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1770 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)(17,33)(18,23)(19,35)(20,26)(21,32)(24,34)(25,41)(27,37)(29,39)(30,36)(31,44)(38,43)(40,42);(1,2)(3,4)(6,10)(11,12)(13,15,18,26)(14,20,23,22)(16,29,31,30)(17,34,27,19)(21,38,40,41)(24,33,35,37)(25,42,43,32)(28,36,44,39);(5,7,9,8)(6,11,10,12)(13,16,19,32,18,31,34,42)(14,21,24,28,23,40,35,44)(15,25,27,29,26,43,17,30)(20,36,37,38,22,39,33,41);(2,4)(13,17,19,15,18,27,34,26)(14,22,24,33,23,20,35,37)(16,30,32,25,31,29,42,43)(21,39,28,41,40,36,44,38);(5,8,9,7)(6,11,10,12)(14,23)(16,31)(20,22)(24,35)(25,43)(29,30)(32,42)(33,37)Polycyclic presentation:
2^9|b^2=i,c^2=fgi,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, 7471419);