Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3585 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C42 · Schur multiplier (H2) = C211 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator | C4 | C23×C42 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
7766 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)(5,6,7,8)(10,15)(12,13)(16,22)(17,18)(20,21)(23,24);(2,4)(9,10,13,18,14,17,12,15)(11,16,21,24,19,23,20,22);(9,11)(10,16)(12,20)(13,21)(14,19)(15,22)(17,23)(18,24);(11,19)(16,23)(20,21)(22,24);(10,17)(15,18)(16,23)(22,24)Polycyclic presentation:
2^9|a^2=g,b^2=h,f^2=i,h^2=i,b^a=bf,f^a=fi,h^a=hi,e^b=ei,d^c=di164 conjugacy classes — view the full character table.
| maximal subgroup of | #72234294, #72260873, #72260912, … |
| maximal normal subgroup of | #72234294, #72260873, #72260912, … |
| maximal quotient of | #72233872 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6276948);