Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24274 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C42 |
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 | C22×#469 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1948 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 20 points:
(5,6)(7,8)(9,10,11,12)(14,17)(15,19)(18,20);(1,2,3,4)(5,6)(7,8)(13,14,15,17,16,18,19,20);(2,4)(6,8)(10,12)(14,18)(17,20);(13,14,19,20,16,18,15,17)Polycyclic presentation:
2^9|a^2=h,b^2=efghi,d^2=ei,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,d^b=di,d^c=di92 conjugacy classes — view the full character table.
| maximal subgroup of | #72264553, #72264562, #72264598 |
| maximal normal subgroup of | #72264553, #72264562, #72264598 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 274255);