Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #34980 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C25 |
| commutator | C23 | C24×C4 |
| prime radical | C22×#538 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
5590 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(5,6,7,8)(9,10)(11,16)(12,14)(13,15)(17,18,20,22)(19,21,23,24);(2,4)(5,6,7,8)(9,11,13,12)(10,14,15,16)(17,18)(19,21)(20,22)(23,24);(1,2)(3,4)(5,6)(7,8)(10,15)(14,16)(19,23)(21,24);(9,10)(11,14)(12,16)(13,15)(17,19)(18,21)(20,23)(22,24);(9,12,13,11)(10,16,15,14)(17,18)(19,21)(20,22)(23,24)Polycyclic presentation:
2^9|a^2=i,b^2=fi,e^2=gh,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^c=df128 conjugacy classes — view the full character table.
| maximal subgroup of | #72335533, #72335542, #72335553 |
| maximal normal subgroup of | #72335533, #72335542, #72335553 |
| maximal quotient of | #72335607 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 423910);