Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24264 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×#135 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3890 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2)(3,4)(5,6)(7,8)(13,14,18,20,17,21,16,22)(15,23,27,28,26,19,25,24);(2,4)(9,10,11,12)(13,15,18,27,17,26,16,25)(14,19,20,24,21,23,22,28);(6,8)(10,12)(13,14)(15,24)(16,20)(17,21)(18,22)(19,27)(23,25)(26,28);(2,4)(13,16,17,18)(14,20,21,22)(15,25,26,27)(19,24,23,28)Polycyclic presentation:
2^9|a^2=h,b^2=gi,d^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,f^c=fi,g^c=gi,h^c=hi116 conjugacy classes — view the full character table.
| maximal subgroup of | #72264595, #72264604, #72264631, … |
| maximal normal subgroup of | #72264595, #72264604, #72264631, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 268789);