Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61071 |
| 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×#3154 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2891 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)(9,10,14,16,15,20,13,19)(11,17,22,12,18,24,21,23);(5,7,8,6)(10,16)(12,17)(13,14)(19,20)(21,22)(23,24);(1,2,3,4)(9,11)(10,17)(12,16)(13,21)(14,22)(15,18)(19,23)(20,24);(9,12,13,17,15,23,14,24)(10,18,19,22,20,11,16,21);(11,18)(12,23)(17,24)(21,22)Polycyclic presentation:
2^9|a^2=h,b^2=fh,c^2=g,d^2=f,f^2=i,h^2=i,b^a=bf,c^a=cg,d^b=df,f^b=fi,h^b=hi,e^c=ei,e^d=ei110 conjugacy classes — view the full character table.
| maximal subgroup of | #72284435, #72284617, #72284953 |
| maximal normal subgroup of | #72284435, #72284617, #72284953 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6289992);