Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437521 |
| 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 | |
5218 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)(9,10,15,18)(11,13,17,14)(12,16,23,20)(19,21,24,22);(1,2,3,4)(9,11,13,18,15,17,14,10)(12,20,22,19,23,16,21,24);(5,6,7,8)(9,12)(10,16)(11,19)(13,21)(14,22)(15,23)(17,24)(18,20);(9,12)(10,16)(11,19)(13,21)(14,22)(15,23)(17,24)(18,20);(9,13,15,14)(10,11,18,17)(12,21,23,22)(16,19,20,24)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=g,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=dh,f^c=fi,h^c=hi,f^d=fi,h^d=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72308676, #72308873, #72308910, … |
| maximal normal subgroup of | #72308676, #72308873, #72308910, … |
| maximal quotient of | #72315682 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6350354);