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 | |
2810 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,4)(5,6)(7,8)(9,10,15,17,14,18,13,19)(11,20,22,23,16,24,21,12);(1,2,3,4)(9,11,13,21,14,16,15,22)(10,12,17,20,18,23,19,24);(6,8)(9,12)(10,16)(11,18)(13,24)(14,23)(15,20)(17,21)(19,22);(11,16)(12,23)(20,24)(21,22)Polycyclic presentation:
2^9|a^2=h,b^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,g^a=gi,c^b=cg,d^b=di,d^c=di,e^c=ei,g^c=gi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72238301, #72240011, #72240013, … |
| maximal normal subgroup of | #72238301, #72240011, #72240013, … |
| maximal quotient of | #72238285, #72240017, #72321836 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265003);