Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #476609 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2749 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
687 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(3,7)(5,8)(9,10,14,19,13,17,22,18)(11,15,24,23,20,12,16,21);(1,2)(3,5)(4,6)(7,8)(9,11)(10,15,17,12)(13,20)(14,16)(18,23,19,21)(22,24);(1,3,4,7)(2,5,6,8)(9,12,13,15)(10,16,17,24)(11,19,20,18)(14,23,22,21);(2,6)(5,8)(10,17)(11,20)(14,22)(21,23)Polycyclic presentation:
2^9|a^2=eg,b^2=h,c^2=ef,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=def,e^b=ei,f^b=fi,d^c=de,g^c=gi,h^c=hi,g^d=gi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 386972);