Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24269 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
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 | |
1350 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(14,19)(15,23,25,26)(20,24,27,28)(21,22);(6,9)(10,12)(13,14,18,22,16,19,17,21)(15,24,26,27,25,28,23,20);(2,4)(5,7,8,11)(6,10,9,12)(13,15,16,25)(14,20,19,27)(17,26,18,23)(21,24,22,28);(6,9)(10,12)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,f^c=fi,g^c=gi,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72317235, #72317237, #72317238, … |
| maximal normal subgroup of | #72317235, #72317237, #72317238, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 269022);