Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60367 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#585 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
17302 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)(17,21)(19,20)(23,24);(9,10)(11,16)(12,15)(13,21)(14,17)(18,23)(19,22)(20,24);(2,4)(9,11)(10,16)(12,19)(13,20)(14,18)(15,22)(17,23)(21,24);(11,18)(16,23)(19,20)(22,24);(9,12,14,13)(10,15,17,21)(11,19,18,20)(16,22,23,24)Polycyclic presentation:
2^9|a^2=h,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,f^b=fi,d^c=di116 conjugacy classes — view the full character table.
| maximal subgroup of | #72239049, #72249569, #72249572, … |
| maximal normal subgroup of | #72239049, #72249569, #72249572, … |
| maximal quotient of | #72239068, #72239092, #72239103, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6277917);