Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806839 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #42927 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2160 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,25)(16,17)(18,22)(19,24)(20,26)(21,23)(27,28);(6,9)(7,11)(13,15,18,26)(14,20,22,25)(16,27,23,19)(17,24,21,28);(1,2,3,4)(13,16,19,15,18,23,27,26)(14,21,24,20,22,17,28,25);(6,9)(10,12)(14,22)(17,21)(20,25)(24,28);(5,7,8,11)(6,10,9,12)(13,17,27,20,18,21,19,25)(14,23,28,15,22,16,24,26)Polycyclic presentation:
2^9|b^2=i,c^2=ghi,e^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6723099);