Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D8×#120 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#120 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2224 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(5,6)(7,11)(8,10)(9,12);(5,7,9,8)(6,10,12,11)(13,14)(15,22)(16,19)(17,21)(18,23)(20,26)(24,28)(25,27);(2,4)(5,7,9,8)(6,11,12,10)(13,15,17,24)(14,18,21,27)(16,25,26,23)(19,22,20,28);(5,7,9,8)(6,11,12,10)(13,14)(15,22)(16,19)(17,21)(18,23)(20,26)(24,28)(25,27);(5,8,9,7)(6,10,12,11)(14,19)(16,26)(18,22)(20,21)(23,25)(27,28)Polycyclic presentation:
2^9|b^2=f,c^2=fi,d^2=f,e^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6486070);