Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437753 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3449 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4766 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)(9,10,15,17,14,19,13,18)(11,20,21,12,16,24,22,23);(5,6)(7,8)(9,11)(10,12)(13,21)(14,16)(15,22)(17,20)(18,24)(19,23);(2,4)(6,8)(9,12)(10,16)(11,19)(13,20)(14,23)(15,24)(17,22)(18,21);(6,8)(11,16)(12,23)(20,24)(21,22);(9,13,14,15)(10,17,19,18)(11,21,16,22)(12,20,23,24)Polycyclic presentation:
2^9|a^2=fi,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,f^b=fi,d^c=di80 conjugacy classes — view the full character table.
| maximal subgroup of | #72315707, #72315753, #72316057, … |
| maximal normal subgroup of | #72315707, #72315753, #72316057, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6344953);