Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61049 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3155 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2980 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(5,6,8,7)(9,10,14,12,15,19,13,18)(11,17,21,20,22,23,24,16);(5,7,8,6)(9,11)(10,16)(12,23)(13,21)(14,24)(15,22)(17,18)(19,20);(1,2,3,4)(9,11)(10,17)(12,20)(13,24)(14,21)(15,22)(16,18)(19,23);(9,12,13,10,15,18,14,19)(11,20,24,17,22,16,21,23);(9,13,15,14)(10,18,19,12)(11,21,22,24)(16,17,20,23)Polycyclic presentation:
2^9|a^2=h,b^2=fhi,c^2=g,d^2=fi,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,d^b=df,f^b=fi,h^b=hi,e^c=ei116 conjugacy classes — view the full character table.
| maximal subgroup of | #72284314, #72284953 |
| maximal normal subgroup of | #72284314, #72284953 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6289979);