Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #346446 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2846 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #62818 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
2576 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,5)(3,7)(6,8)(9,10,13,16)(11,14,18,21)(12,15,19,22)(17,23,24,20);(1,2)(3,8)(4,6)(5,7)(9,11)(10,14)(12,17)(13,18)(15,20)(16,21)(19,24)(22,23);(9,12,13,19)(10,15,16,22)(11,17,18,24)(14,20,21,23);(3,7)(5,8)Polycyclic presentation:
2^9|a^2=i,c^2=i,e^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,e^b=eh,f^b=fi,g^b=gh,e^c=ei,e^d=eh59 conjugacy classes — view the full character table.
| maximal subgroup of | #72214220, #72233005, #72233013, … |
| maximal normal subgroup of | #72214220, #72233005, #72233013, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 253666);