Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #181344 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#95 |
| commutator | C23 | C22×C42 |
| Frattini, cosocle | C25 | C24 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C23 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×D4 ◁ G | |
| LECS | G ▷ C25 ▷ C22 ▷ C1 | |
3268 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,17,19,14)(12,15)(13,16)(18,23,24,21)(20,22);(1,3)(2,5,6,7)(4,8)(9,11)(10,14)(12,18)(13,19)(15,21)(16,17)(20,24)(22,23);(9,12)(10,15)(11,18)(13,20)(14,21)(16,22)(17,23)(19,24);(12,20)(15,22)(18,24)(21,23)Polycyclic presentation:
2^9|a^2=f,b^2=g,b^a=be,e^a=eh,g^a=gi,e^b=ei,f^b=fh,d^c=dh92 conjugacy classes — view the full character table.
| maximal subgroup of | #72232871, #72233182, #72233199, … |
| maximal normal subgroup of | #72232871, #72233182, #72233199, … |
| maximal quotient of | #72232856, #72233041, #72236734, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 88540);