Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437758 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3512 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3032 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6,8,9)(7,10,11,12)(13,14,19,23)(15,24,28,39)(16,29,17,21)(18,32,33,22)(20,34,36,27)(25,40,26,38)(30,37,31,44)(35,41,43,42);(1,2)(3,4)(13,15,18,27,19,28,33,34)(14,20,22,24,23,36,32,39)(16,25,30,41,17,26,31,42)(21,35,37,38,29,43,44,40);(2,4)(6,9)(10,12)(13,16,19,17)(14,21,23,29)(15,25,28,26)(18,30,33,31)(20,35,36,43)(22,37,32,44)(24,38,39,40)(27,41,34,42);(2,4)(5,7,8,11)(6,10,9,12)(13,17,19,16)(14,21,23,29)(15,26,28,25)(18,31,33,30)(20,35,36,43)(22,37,32,44)(24,38,39,40)(27,42,34,41);(16,17)(21,29)(25,26)(30,31)(35,43)(37,44)(38,40)(41,42)Polycyclic presentation:
2^9|a^2=g,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=dh,e^c=ei,e^d=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6357271);