Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35346 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | #34736 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4748 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)(5,6)(7,8)(9,10,14,17,13,12,16,21)(11,15,20,23,19,18,22,24);(2,4)(9,11)(10,15)(12,18)(13,19)(14,20)(16,22)(17,23)(21,24);(6,8)(10,12)(15,18)(17,21)(23,24);(9,12,14,21,13,10,16,17)(11,18,20,24,19,15,22,23);(11,19)(15,18)(20,22)(23,24)Polycyclic presentation:
2^9|a^2=h,d^2=h,h^2=i,b^a=bf,c^a=cg,e^b=ei,d^c=di128 conjugacy classes — view the full character table.
| maximal subgroup of | #72249635, #72249641, #72256868, … |
| maximal normal subgroup of | #72249635, #72249641, #72256868, … |
| maximal quotient of | #72257850, #72264638, #72307967, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6279803);