Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3055 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C23 | C23×D4 |
| prime radical | C22×#575 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
5366 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(5,6,7,8)(9,10)(11,21)(12,24)(13,18)(14,19)(15,20)(16,23)(17,22);(1,2,3,4)(6,8)(9,11,13,23,15,22,14,12)(10,16,19,21,20,24,18,17);(9,12,14,22,15,23,13,11)(10,17,18,24,20,21,19,16);(11,22)(12,23)(16,24)(17,21);(9,13,15,14)(10,18,20,19)(11,23,22,12)(16,17,24,21)Polycyclic presentation:
2^9|a^2=h,b^2=fh,c^2=g,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,d^b=di,d^c=di128 conjugacy classes — view the full character table.
| maximal subgroup of | #72224731, #72224734, C4×#60639, … |
| maximal normal subgroup of | #72224731, #72224734, C4×#60639, … |
| maximal quotient of | #72224713, #72224715, #72264920, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6279198);