Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #686200 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #15310 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3144 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)(6,9)(10,12)(13,14)(15,17)(16,18)(19,20)(22,26)(23,27);(13,14,16,18)(15,17,19,20)(21,22,25,26)(23,24,27,28);(1,2,3,4)(15,19)(17,20)(22,26)(23,27);(5,6)(7,10)(8,9)(11,12)(13,15,16,19)(14,17,18,20)(21,23)(22,24)(25,27)(26,28);(5,7,8,11)(6,10,9,12)(13,14)(15,17)(16,18)(19,20)(21,24)(22,23)(25,28)(26,27)Polycyclic presentation:
2^9|b^2=f,c^2=g,d^2=fh,e^2=hi,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,d^c=df92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 658814);