Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10374 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #393 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C22 | #1439 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
858 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,4,7,5,3,8,6)(9,10)(11,14)(12,19,21,15)(13,16)(17,24,20,22)(18,23);(9,11)(10,14)(12,20,21,17)(13,18)(15,22,19,24)(16,23);(1,3)(2,5)(4,6)(7,8)(9,12)(10,15)(11,17)(13,21)(14,22)(16,19)(18,20)(23,24)Polycyclic presentation:
2^9|a^2=g,b^2=d,e^2=hi,g^2=hi,b^a=bd,c^a=ce,e^a=ei,f^a=fh,c^b=cf,f^b=fh,d^c=dh,e^c=ehi,g^c=gh74 conjugacy classes — view the full character table.
| maximal subgroup of | #72266586, #72266644 |
| maximal normal subgroup of | #72266586, #72266644 |
| maximal quotient of | #72266592 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 23288);