Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27 |
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 | |
2168 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)(5,6,9,8)(7,10,11,12)(14,17)(18,19)(21,22,25,27)(23,24,28,26);(6,8)(10,12)(13,14)(15,19)(16,17)(18,20)(21,23,25,28)(22,26,27,24);(7,11)(10,12)(14,17)(18,19)(22,27)(23,28);(5,7,9,11)(6,10,8,12)(13,14)(15,18)(16,17)(19,20)(21,23)(22,26)(24,27)(25,28);(5,8,9,6)(7,12,11,10)(13,15)(14,18)(16,20)(17,19)(21,24,25,26)(22,23,27,28)Polycyclic presentation:
2^9|a^2=i,b^2=g,d^2=fh,e^2=fgh,b^a=bf,c^a=cg,d^a=dh,e^a=eg,c^b=ch,d^b=dg,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, 494175);