Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C2×D4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C24 | C22×D4 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
4586 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2)(3,4)(5,6,7,8)(13,14)(15,24)(16,21)(17,22)(18,23)(19,26)(20,25)(27,28);(2,4)(9,10,11,12)(13,15)(14,19)(16,25)(17,26)(18,27)(20,23)(21,24)(22,28);(13,15)(14,20)(16,25)(17,26)(18,27)(19,23)(21,28)(22,24);(13,16,18,17)(14,21,23,22)(15,25,27,26)(19,24,20,28)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=i,e^2=i,f^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,e^b=ei,f^b=fi,e^c=ei,f^c=fi152 conjugacy classes — view the full character table.
| maximal subgroup of | #72315984, #72315987, #72316009, … |
| maximal normal subgroup of | #72315984, #72315987, #72316009, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 260562);