Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437764 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3400 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6)(7,10)(8,9)(11,12)(13,14)(15,24)(16,28)(17,21)(18,32)(19,23)(20,26)(22,33)(25,39)(27,40)(29,37)(30,38)(31,43)(34,44)(35,41)(36,42);(1,2)(3,4)(13,15,18,26,19,27,33,42)(14,20,22,24,23,36,32,40)(16,25,29,41,17,30,31,44)(21,34,37,38,28,35,43,39);(2,4)(6,9)(10,12)(13,16,19,17)(14,21,23,28)(15,25,27,30)(18,29,33,31)(20,34,36,35)(22,37,32,43)(24,38,40,39)(26,41,42,44);(2,4)(5,7,8,11)(6,10,9,12)(13,17,19,16)(14,21,23,28)(15,25,27,30)(18,31,33,29)(20,35,36,34)(22,37,32,43)(24,39,40,38)(26,41,42,44);(16,17)(21,28)(25,30)(29,31)(34,35)(37,43)(38,39)(41,44)Polycyclic presentation:
2^9|b^2=f,c^2=i,d^2=g,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dhi,e^c=ei,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6356360);