Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438975 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×2−1+4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2692 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,10)(8,9)(11,12)(13,14,20,24)(15,25,16,28)(17,22,21,27)(18,23,19,26);(1,2)(3,4)(6,9)(10,12)(13,15,19,22,20,16,18,27)(14,21,23,25,24,17,26,28);(2,4)(5,7,8,11)(6,10,9,12)(13,16,19,27,20,15,18,22)(14,17,23,28,24,21,26,25);(2,4)(13,17)(14,22)(15,26)(16,23)(18,25)(19,28)(20,21)(24,27);(13,18,20,19)(14,23,24,26)(15,27,16,22)(17,28,21,25)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=f,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=dh,e^d=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6347991);