Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23387 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
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 | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×M16 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2658 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)(9,10,13,18,15,11,14,17)(12,20,23,19,22,21,24,16);(5,6)(7,8)(9,11)(10,15)(12,21)(13,17)(14,18)(16,23)(19,24)(20,22);(2,4)(6,8)(9,12,13,23,15,22,14,24)(10,16,17,21,11,19,18,20);(12,22)(16,19)(20,21)(23,24)Polycyclic presentation:
2^9|a^2=h,c^2=e,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,d^c=di104 conjugacy classes — view the full character table.
| maximal subgroup of | #72238301, #72238813, #72239937 |
| maximal normal subgroup of | #72238301, #72238813, #72239937 |
| maximal quotient of | #72238285, #72238820, #72239947 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265155);