Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12077 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #442 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C23×D8 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
852 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,4)(5,6,7,8)(10,16,19,17)(11,21)(12,22)(13,14)(18,23,20,24);(9,10,13,20,15,19,14,18)(11,17,21,24,22,16,12,23);(1,2,3,4)(6,8)(9,11,14,12,15,22,13,21)(10,17,18,23,19,16,20,24)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=efi,d^2=h,e^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,g^a=gi,h^a=hi,d^b=dh,e^b=ei,f^b=fi,h^b=hi,d^c=di80 conjugacy classes — view the full character table.
| maximal subgroup of | #72309654, #72309659, #72309672, … |
| maximal normal subgroup of | #72309654, #72309659, #72309672, … |
| maximal quotient of | #72322463, #72322523 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54661);