Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61058 |
| 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×#3061 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2690 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,4)(5,6,8,7)(9,10,15,16,14,19,13,20)(11,21,23,24,18,17,22,12);(5,7,8,6)(10,16)(12,17)(13,15)(19,20)(21,24)(22,23);(2,4)(9,11)(10,17)(12,16)(13,22)(14,18)(15,23)(19,21)(20,24);(9,12,13,17,14,24,15,21)(10,18,20,23,19,11,16,22);(10,19)(11,18)(16,20)(22,23)Polycyclic presentation:
2^9|a^2=h,b^2=fh,d^2=f,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,d^b=df,f^b=fi,h^b=hi,e^c=ei104 conjugacy classes — view the full character table.
| maximal subgroup of | #72320617, #72320707, #72320750, … |
| maximal normal subgroup of | #72320617, #72320707, #72320750, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6289168);