Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #647656 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
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 | |
4514 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)(10,13)(14,16)(17,18,21,23)(19,20,24,22);(6,8)(17,19,21,24)(18,22,23,20);(5,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,20)(18,19)(21,22)(23,24);(1,2)(3,4)(9,10)(11,14)(12,13)(15,16)(17,20)(18,19)(21,22)(23,24);(9,11,12,15)(10,14,13,16)(17,19)(18,20)(21,24)(22,23)Polycyclic presentation:
2^9|a^2=f,b^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef104 conjugacy classes — view the full character table.
| maximal subgroup of | #72280809, #72280877, #72280940, … |
| maximal normal subgroup of | #72280809, #72280877, #72280940, … |
| maximal quotient of | #72323481, C2×#647925, #72331302, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 555121);