Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #52308 |
| 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 | C22×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2480 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,4,3)(6,7)(8,12)(14,16)(18,20)(22,25)(23,27);(1,3,4,2)(5,6,9,7)(8,10,12,11)(13,14,17,16)(15,18,19,20)(21,22,24,25)(23,26,27,28);(6,7)(10,11)(15,19)(18,20)(21,22,24,25)(23,26,27,28);(5,7,9,6)(8,11,12,10)(13,15,17,19)(14,18,16,20)(21,22)(23,26)(24,25)(27,28);(5,8,9,12)(6,10,7,11)(13,16,17,14)(15,20,19,18)(21,23,24,27)(22,26,25,28)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=g,d^2=fg,e^2=f,b^a=bf,c^a=cg,d^a=dgh,e^a=ef,c^b=ch,d^b=dg,d^c=df92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 489598);