Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53158 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C25 |
| commutator | C23 | C24×C4 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2222 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)(5,6)(7,11)(8,9)(10,12)(14,17)(15,19)(22,24)(26,28);(6,8)(10,11)(14,17)(18,20)(23,27)(26,28);(13,14)(15,18)(16,17)(19,20)(21,22,25,24)(23,26,27,28);(5,7,9,12)(6,10,8,11)(13,14)(15,18)(16,17)(19,20)(21,23)(22,26)(24,28)(25,27);(5,8,9,6)(7,11,12,10)(13,15,16,19)(14,18,17,20)(21,24,25,22)(23,28,27,26)Polycyclic presentation:
2^9|a^2=i,c^2=gh,d^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=dfh,e^a=efg,c^b=ch,d^b=dg,e^b=ef92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 514015);