Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24269 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C25 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C22⋊C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1378 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,4)(9,10,14,20,15,19,22,18)(11,21,12,16,13,24,23,17);(5,6)(7,8)(9,11,14,12,15,13,22,23)(10,16,18,21,19,17,20,24);(2,4)(6,8)(9,12,14,13,15,23,22,11)(10,16,20,24,19,17,18,21);(9,13,14,23,15,11,22,12)(10,17,18,24,19,16,20,21)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=h,d^2=e,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,d^c=di92 conjugacy classes — view the full character table.
| maximal subgroup of | #72250015, #72250020, #72250022, … |
| maximal normal subgroup of | #72250015, #72250020, #72250022, … |
| maximal quotient of | #72265410, C2×#373855, #72322476 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 281051);