Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3484 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C42 · Schur multiplier (H2) = C211 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator | C22 | C23×C42 |
| socle | C23 | C24×C4 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
7854 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)(9,10,12,16,13,15,19,22)(11,14,18,21,17,20,23,24);(2,4)(5,6,7,8);(9,11)(10,14)(12,18)(13,17)(15,20)(16,21)(19,23)(22,24);(11,17)(14,20)(18,23)(21,24);(10,15)(14,20)(16,22)(21,24)Polycyclic presentation:
2^9|a^2=g,b^2=h,g^2=i,b^a=bf,e^a=ei,d^c=di176 conjugacy classes — view the full character table.
| maximal subgroup of | #72279388, #72279403, #72279440, … |
| maximal normal subgroup of | #72279388, #72279403, #72279440, … |
| maximal quotient of | #72280217, C2×#6369736, #72303584 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6276932);