Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #500834 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C24 | C2×D8 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊C4 ◁ C8×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
562 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6,8,10)(7,11,12,9)(14,19)(15,23)(16,21)(18,25)(20,26)(22,28)(24,27);(1,2)(3,4)(13,14,20,28,18,22,16,19,17,15,21,27,25,24,26,23);(2,4)(5,7,8,12)(6,9,10,11)(14,15)(19,23)(22,24)(27,28);(13,15,20,27,18,24,16,23,17,14,21,28,25,22,26,19)Polycyclic presentation:
2^9|a^2=f,b^2=ehi,c^2=f,d^2=ehi,e^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg,d^c=di74 conjugacy classes — view the full character table.
| maximal subgroup of | #72319014, #72319556, #72319638 |
| maximal normal subgroup of | #72319014, #72319556, #72319638 |
| maximal quotient of | #72322632 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 410957);