Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C25 |
| commutator | C23 | C24×C4 |
| prime radical | C22×#392 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
6102 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(5,6,7,8)(9,10)(11,15)(12,13)(14,16)(18,21)(22,24);(2,4)(10,13)(14,15)(17,18)(19,22)(20,21)(23,24);(1,2,3,4)(10,13)(14,15)(19,23)(22,24);(9,10)(11,14)(12,13)(15,16)(17,19)(18,22)(20,23)(21,24);(9,11)(10,14)(12,16)(13,15)(17,18)(19,22)(20,21)(23,24)Polycyclic presentation:
2^9|a^2=i,c^2=h,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dg,d^c=df104 conjugacy classes — view the full character table.
| maximal subgroup of | #72232503, #72232504, #72232569, … |
| maximal normal subgroup of | #72232503, #72232504, #72232569, … |
| maximal quotient of | #72323684, #72323685, C2×#544502, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 451698);