Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5955303 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C22×#120 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
4408 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)(10,13)(11,15)(17,18,22,23)(19,20,24,21);(2,4)(10,13)(14,16)(18,23)(19,24);(1,2)(3,4)(5,6)(7,8)(17,19)(18,20)(21,23)(22,24);(9,10)(11,14)(12,13)(15,16)(17,20,22,21)(18,24,23,19);(9,11)(10,14)(12,15)(13,16)(17,21,22,20)(18,19,23,24)Polycyclic presentation:
2^9|a^2=fg,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ei,c^b=ch,d^b=di104 conjugacy classes — view the full character table.
| maximal subgroup of | #72332306, #72332313, #72332328 |
| maximal normal subgroup of | #72332306, #72332313, #72332328 |
| maximal quotient of | #72333139 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5862515);