Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453991 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2669 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23428 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1217 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)(11,19)(12,15)(13,18)(14,17)(16,20)(21,23)(22,24);(1,2,4,5)(3,6,7,8)(9,11)(10,16)(12,23)(13,21)(14,20)(15,22)(17,24)(18,19);(1,3,4,7)(2,6,5,8)(9,12,13,18,15,10,14,17)(11,16,20,19,22,23,21,24);(3,7)(6,8)(9,10)(11,16)(12,15)(13,18)(14,17)(19,21)(20,24)(22,23)Polycyclic presentation:
2^9|b^2=ef,c^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,f^b=fi,h^b=hi,d^c=de,g^c=gi,e^d=ei,f^d=fi,g^d=gi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 363162);