Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61049 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2298 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1988 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)(7,10)(8,9)(11,12)(13,14)(15,23)(16,19)(17,22)(18,27)(20,24)(21,25)(26,28);(13,15,19,25)(14,20,16,26)(17,24,18,28)(21,27,23,22);(6,9)(10,12)(13,15)(14,21)(16,23)(17,28)(18,24)(19,25)(20,22)(26,27);(13,16,19,14)(15,23,25,21)(17,27,18,22)(20,28,26,24);(5,7,8,11)(6,10,9,12)(13,17,19,18)(14,22,16,27)(15,24,25,28)(20,21,26,23)Polycyclic presentation:
2^9|a^2=h,b^2=i,d^2=i,e^2=g,f^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fi,c^b=ci,d^b=df,f^b=fi,e^c=ei,f^d=fi116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6288163);