Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437537 |
| 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×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#533 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3072 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6,8,9)(7,10,11,12)(13,14)(15,23)(16,20)(17,21)(18,22)(19,25)(24,27)(26,28);(1,2,3,4)(13,14)(15,20)(16,23)(17,21)(18,27)(19,26)(22,24)(25,28);(6,9)(10,12)(13,15,17,25)(14,19,21,23)(16,24,26,18)(20,22,28,27);(5,7,8,11)(6,10,9,12)(13,16)(14,20)(15,24)(17,26)(18,25)(19,22)(21,28)(23,27);(14,21)(19,23)(20,28)(22,27)Polycyclic presentation:
2^9|a^2=gi,b^2=f,c^2=i,d^2=gi,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,e^b=ei,h^b=hi,h^c=hi,h^d=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6357196);