Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60454 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28×C4 |
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 | C23×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3386 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,4)(5,6,8,7)(9,10)(11,20)(12,22)(13,18)(14,24)(15,19)(16,17)(21,23);(5,7,8,6)(9,11,14,17,15,22,13,21)(10,16,18,20,19,23,24,12);(2,4)(9,12,14,16,15,20,13,23)(10,17,18,22,19,21,24,11);(9,13,15,14)(10,18,19,24)(11,21,22,17)(12,16,20,23);(9,13,15,14)(10,18,19,24)(11,17,22,21)(12,23,20,16)Polycyclic presentation:
2^9|a^2=h,b^2=fh,c^2=f,d^2=i,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,e^b=ei,d^c=di116 conjugacy classes — view the full character table.
| maximal subgroup of | #72308796, #72308820, #72308847, … |
| maximal normal subgroup of | #72308796, #72308820, #72308847, … |
| maximal quotient of | #72316211 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6278893);