Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437764 |
| 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×#535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3108 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6,8,9)(7,10,11,12)(13,14,20,23)(15,18,22,19)(16,25,17,21)(24,27,26,28);(1,2)(3,4)(13,15,19,14,20,22,18,23)(16,26,27,25,17,24,28,21);(2,4)(6,9)(10,12)(13,16,20,17)(14,21,23,25)(15,24,22,26)(18,28,19,27);(2,4)(5,7,8,11)(6,10,9,12)(13,17,20,16)(14,21,23,25)(15,24,22,26)(18,27,19,28);(13,18,20,19)(14,22,23,15)(16,27,17,28)(21,24,25,26)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=i,d^2=g,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dhi,e^b=ei,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, 6357289);