Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #267371 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2410 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #61559 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1245 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)(6,8)(9,10)(11,17,19,14)(12,20)(13,16)(15,21)(18,23,24,22);(2,6)(4,7)(5,8)(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,23)(21,24);(1,2)(3,5)(4,8)(6,7)(9,12)(10,15)(11,18)(13,21)(14,22)(16,20)(17,23)(19,24);(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,23)(21,24)Polycyclic presentation:
2^9|a^2=fhi,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,f^b=fh,g^b=gi,e^c=eh,g^c=gi,f^d=fh59 conjugacy classes — view the full character table.
| maximal subgroup of | #72268297, #72268307, #72268311, … |
| maximal normal subgroup of | #72268297, #72268307, #72268311, … |
| maximal quotient of | #72313309 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 174572);