Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60460 |
| 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 | C22×#469 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3010 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)(11,18)(12,16)(13,22)(14,17)(15,21)(19,24)(20,23);(9,11,12,20,14,19,13,21)(10,15,22,24,17,23,16,18);(2,4)(11,19)(15,23)(18,24)(20,21);(5,7,8,6)(9,10)(11,15)(12,22)(13,16)(14,17)(18,21)(19,23)(20,24);(9,12,14,13)(10,16,17,22)(11,20,19,21)(15,18,23,24)Polycyclic presentation:
2^9|a^2=h,b^2=fi,d^2=h,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ci,e^d=ei116 conjugacy classes — view the full character table.
| maximal subgroup of | #72224708, #72226672, #72226674, … |
| maximal normal subgroup of | #72224708, #72226672, #72226674, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6280186);