Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437547 |
| 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 | #35513 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2898 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)(7,10)(8,9)(11,12)(13,14)(15,21)(16,19)(17,26)(18,24)(20,25)(22,27)(23,28);(1,2)(3,4)(6,9)(10,12)(13,14,16,19)(15,22,23,18)(17,20,25,26)(21,24,28,27);(2,4)(5,7,8,11)(6,10,9,12)(13,15,16,23)(14,18,19,22)(17,27,25,24)(20,28,26,21);(2,4)(13,14)(15,18)(16,19)(17,20)(21,24)(22,23)(25,26)(27,28);(14,19)(18,22)(20,26)(21,28)Polycyclic presentation:
2^9|b^2=i,c^2=f,g^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,c^b=ch,d^b=dh,e^b=ei,g^c=gi,e^d=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6345556);