Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437529 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
5306 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)(6,8)(9,10,16,19)(11,14,18,15)(12,21,13,17)(20,23,24,22);(1,2,3,4)(9,11,15,10,16,18,14,19)(12,21,22,24,13,17,23,20);(5,6,7,8)(9,12)(10,17)(11,20)(13,16)(14,22)(15,23)(18,24)(19,21);(9,13)(10,17)(11,20)(12,16)(14,23)(15,22)(18,24)(19,21);(9,14,16,15)(10,11,19,18)(12,22,13,23)(17,20,21,24)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=gi,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,h^a=hi,c^b=ch,d^b=dhi,f^c=fi,h^c=hi,f^d=fi,h^d=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72283430, #72308686, #72308698, … |
| maximal normal subgroup of | #72283430, #72308686, #72308698, … |
| maximal quotient of | #72315779, #72315873 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6350363);