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 | C2×#392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2304 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)(5,6,8,9)(7,11,12,10)(13,14,16,18)(15,17,19,20)(22,27)(23,25);(1,3,4,2)(6,9)(10,11)(13,14)(15,17)(16,18)(19,20)(21,22,26,27)(23,24,25,28);(6,9)(10,11)(15,19)(17,20)(21,23,26,25)(22,28,27,24);(5,6)(7,10)(8,9)(11,12)(13,15)(14,17)(16,19)(18,20)(21,24,26,28)(22,23,27,25);(5,7,8,12)(6,10,9,11)(13,14)(15,17)(16,18)(19,20)(21,25,26,23)(22,28,27,24)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=h,d^2=h,e^2=g,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, 489732);