Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3019 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#127 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2316 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,3)(9,10,13,15)(11,17,19,24)(12,20,22,16)(14,21,23,18);(5,6,7,8)(9,11)(10,14)(12,21)(13,19)(15,23)(16,24)(17,20)(18,22);(12,22)(16,20)(17,24)(18,21);(1,3,4,2)(10,15)(14,23)(16,20)(17,24)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=gi,e^2=i,b^a=be,c^a=cf,d^a=di,e^a=ei,e^b=ei,f^b=fi,e^c=ei116 conjugacy classes — view the full character table.
| maximal subgroup of | #72261845, #72261896, C4×#21473 |
| maximal normal subgroup of | #72261845, #72261896, C4×#21473 |
| maximal quotient of | #72279458 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 261427);