Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2002 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #120 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C2×#120 |
| prime radical | C22×#532 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
4774 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,4)(5,6,7,8)(9,10)(11,16)(12,17)(13,21)(14,18)(15,19)(20,23)(22,24);(9,11)(10,15)(12,19)(13,22)(14,20)(16,21)(17,23)(18,24);(2,4)(9,11)(10,16)(12,19)(13,22)(14,20)(15,17)(18,23)(21,24);(9,12,14,13)(10,17,18,21)(11,19,20,22)(15,23,24,16)Polycyclic presentation:
2^9|a^2=h,d^2=i,e^2=i,b^a=be,c^a=cf,e^a=ei,g^a=gi,c^b=cg,e^b=ei,e^c=ei104 conjugacy classes — view the full character table.
| maximal subgroup of | #72240101, #72258233, #72258236, … |
| maximal normal subgroup of | #72240101, #72258233, #72258236, … |
| maximal quotient of | #72235046, #72267252, #72267268 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 264647);