Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417778 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2912 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,4)(9,10)(11,19)(12,15)(13,18)(14,17)(16,21)(20,23)(22,24);(1,2,3,4)(5,6)(7,8)(9,10,14,18,15,12,13,17)(11,16,23,24,22,20,21,19);(6,8)(9,11)(10,16)(12,20)(13,21)(14,23)(15,22)(17,19)(18,24);(2,4)(9,12,15,10)(11,20,22,16)(13,17,14,18)(19,23,24,21)Polycyclic presentation:
2^9|b^2=ei,d^2=i,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=de,f^c=fi,h^c=hi,e^d=ei,f^d=fi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 324976);