Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535823 |
| 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 | #39462 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2048 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(13,14,20,23)(15,16,26,28)(17,21,27,25)(18,22,19,24);(5,6)(7,10)(8,9)(11,12)(13,15,19,25,20,26,18,21)(14,17,22,16,23,27,24,28);(2,4)(6,9)(10,12)(13,16,20,28)(14,15,23,26)(17,19,27,18)(21,22,25,24);(5,7)(6,10)(8,11)(9,12)(13,17,20,27)(14,21,23,25)(15,24,26,22)(16,18,28,19);(7,11)(10,12)(13,18,20,19)(14,22,23,24)(15,21,26,25)(16,27,28,17)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=i,d^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=di,d^c=di,e^c=ei,e^d=ehi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6452512);