Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #14715 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C23×C4 | C24 |
| commutator, socle | C24 | C23×C4 |
| prime radical | C22×#525 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
4794 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2)(3,4)(5,6)(7,8)(13,14,16,19,17,20,23,26)(15,18,22,25,21,24,27,28);(2,4)(9,10)(11,12);(6,8)(10,12)(13,15)(14,18)(16,22)(17,21)(19,25)(20,24)(23,27)(26,28);(2,4)(15,21)(18,24)(22,27)(25,28)Polycyclic presentation:
2^9|a^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,d^c=di128 conjugacy classes — view the full character table.
| maximal subgroup of | #72320306, #72320318, #72320506 |
| maximal normal subgroup of | #72320306, #72320318, #72320506 |
| maximal quotient of | #72321250 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266619);