Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35346 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | C22×#525 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
5826 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)(6,8)(9,10,13,17,14,11,15,19)(12,16,21,23,20,18,22,24);(1,2,3,4)(9,11,13,19,14,10,15,17)(12,18,21,24,20,16,22,23);(5,6,7,8)(9,12)(10,16)(11,18)(13,21)(14,20)(15,22)(17,23)(19,24);(12,20)(16,18)(21,22)(23,24);(10,11)(16,18)(17,19)(23,24)Polycyclic presentation:
2^9|a^2=h,b^2=fh,c^2=g,h^2=i,b^a=bf,c^a=cg,e^a=ei,e^b=ei,d^c=di128 conjugacy classes — view the full character table.
| maximal subgroup of | #72224722, #72239412, #72249651, … |
| maximal normal subgroup of | #72224722, #72239412, #72249651, … |
| maximal quotient of | #72239404, #72239410, #72305884, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6279128);