Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437753 |
| 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 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
5784 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(10,14)(12,19)(15,20)(16,21)(18,23)(22,24);(5,6)(7,8)(9,10,12,14,13,16,19,21)(11,15,18,20,17,22,23,24);(1,2,3,4)(6,8)(9,11)(10,15)(12,18)(13,17)(14,20)(16,22)(19,23)(21,24);(6,8)(11,17)(15,22)(18,23)(20,24);(10,16)(14,21)(15,22)(20,24)Polycyclic presentation:
2^9|b^2=f,c^2=g,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,e^b=ei,d^c=di80 conjugacy classes — view the full character table.
| maximal subgroup of | #72226694, #72282511, #72282644, … |
| maximal normal subgroup of | #72226694, #72282511, #72282644, … |
| maximal quotient of | #72214731, #72214806 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6350299);