Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #647656 |
| automorphism | ? |
| 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 | #15257 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3988 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)(6,8)(9,10,14,13)(11,16,15,12)(17,18,22,21)(19,20,24,23);(2,4)(18,21)(19,24);(9,11)(10,12)(13,16)(14,15)(17,19,22,24)(18,23,21,20);(5,6)(7,8)(9,12,14,16)(10,15,13,11)(17,20)(18,19)(21,24)(22,23);(1,2)(3,4)(9,13,14,10)(11,16,15,12)(17,21,22,18)(19,20,24,23)Polycyclic presentation:
2^9|a^2=g,c^2=h,d^2=gh,e^2=g,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 | #72232214, #72232220, #72232222, … |
| maximal normal subgroup of | #72232214, #72232220, #72232222, … |
| maximal quotient of | #72298507, #72330943, #72331592, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 559088);