Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #480662 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#99 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C22×D4 | C24 |
| prime radical | #35076 | C2 |
| LCS | G ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1252 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,5)(3,7)(9,10,13,16,12,14,20,21)(11,17,19,24,18,15,22,23);(1,2)(3,6)(4,5)(7,8)(10,14)(11,18)(16,21)(19,22);(2,5)(6,8)(9,11)(10,15)(12,18)(13,19)(14,17)(16,24)(20,22)(21,23);(1,3,4,7)(2,6,5,8)(10,14)(11,18)(16,21)(19,22)Polycyclic presentation:
2^9|a^2=g,d^2=ei,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=ce,d^c=di,g^c=gh,g^f=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72236583, #72240280, #72312499, … |
| maximal normal subgroup of | #72236583, #72240280, #72312499, … |
| maximal quotient of | #72317673 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 403719);