Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | #52096 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2190 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6)(7,10)(8,9)(11,12)(13,14,20,22,19,24,17,26)(15,21,23,27,16,18,25,28);(6,9)(7,11)(13,15)(14,18)(16,19)(17,25)(20,23)(21,24)(22,28)(26,27);(1,2,3,4)(13,16,19,15)(14,21,24,18)(17,23,20,25)(22,27,26,28);(5,7,8,11)(6,10,9,12)(13,17,19,20)(14,22,24,26)(15,23,16,25)(18,27,21,28);(6,9)(10,12)(13,18,20,28,19,21,17,27)(14,23,22,16,24,25,26,15)Polycyclic presentation:
2^9|a^2=h,c^2=g,d^2=f,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ci,d^b=df,d^c=di,e^c=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6311416);