Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24289 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#453 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2493 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)(11,13)(12,21)(14,19)(15,18)(16,23)(17,22)(20,24);(1,2)(3,4)(9,11,13,19,14,18,15,10)(12,21,24,20,23,16,22,17);(2,4)(6,8)(9,12)(10,16)(11,20)(13,22)(14,23)(15,24)(17,18)(19,21);(9,12)(10,17)(11,21)(13,24)(14,23)(15,22)(16,18)(19,20)Polycyclic presentation:
2^9|a^2=h,b^2=fgh,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,e^b=ei,g^c=gi,e^d=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72224689, #72312189, #72312195 |
| maximal normal subgroup of | #72224689, #72312189, #72312195 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 273889);