Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #476297 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2688 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23430 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1153 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,7)(3,6,8,5)(9,10,13,16,14,17,12,18)(11,19,21,24,22,15,20,23);(1,3)(2,5)(4,8)(6,7)(9,11)(10,15)(12,21)(13,20)(14,22)(16,24)(17,19)(18,23);(1,3)(2,6)(4,8)(5,7)(10,16)(11,20)(12,13)(17,18)(21,22)(23,24);(3,8)(5,6)(10,17)(11,21)(12,13)(15,23)(19,24)(20,22)Polycyclic presentation:
2^9|a^2=eg,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=def,f^b=fi,g^b=gi,h^b=hi,d^c=de,f^c=fi,g^c=gi,f^d=fi,g^d=gi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72338919, #72338922 |
| maximal normal subgroup of | #72338919, #72338922 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 385476);