Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3003 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1856 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,4)(9,10,13,17,14,11,18,16)(12,15,20,23,21,19,24,22);(2,4)(5,6,7,8)(9,11)(10,14)(12,15)(13,16)(17,18)(19,21)(20,23)(22,24);(9,10)(11,14)(12,19)(13,17)(15,21)(16,18)(20,22)(23,24);(9,12,13,20,14,21,18,24)(10,15,17,23,11,19,16,22)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=g,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,d^b=di,e^b=ei,f^b=fi,d^c=di,e^c=ei,f^c=fi128 conjugacy classes — view the full character table.
| maximal subgroup of | #72320931, #72320952 |
| maximal normal subgroup of | #72320931, #72320952 |
| maximal quotient of | #72321323, #72321451 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 261522);