Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23387 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
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 | C22×#575 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×M16 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
4490 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)(9,10,13,18,15,11,14,17)(12,16,22,20,23,19,21,24);(2,4)(9,11)(10,15)(12,20)(13,17)(14,18)(16,21)(19,22)(23,24);(6,8)(9,12)(10,16)(11,19)(13,22)(14,21)(15,23)(17,24)(18,20);(10,11)(16,19)(17,18)(20,24)Polycyclic presentation:
2^9|a^2=h,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=di,e^a=ei,g^a=gi,c^b=cg,d^b=di,e^b=ei,g^b=gi,e^c=ei,g^c=gi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72238304, #72320968, #72320984, … |
| maximal normal subgroup of | #72238304, #72320968, #72320984, … |
| maximal quotient of | #72238285, #72321784 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 264718);