Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30412 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×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×D8 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1642 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,3)(9,10,11,12)(14,17)(15,19)(18,20);(6,8)(13,14)(15,20)(16,18)(17,19);(1,2,4,3)(5,6,7,8)(14,18)(17,20);(1,3,4,2)(13,14,19,20,16,18,15,17)Polycyclic presentation:
2^9|a^2=h,c^2=fgi,d^2=efi,e^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,c^b=cg,d^b=de,e^b=ei,d^c=di80 conjugacy classes — view the full character table.
| maximal subgroup of | #72319905, #72320096, #72320263 |
| maximal normal subgroup of | #72319905, #72320096, #72320263 |
| maximal quotient of | #72321498 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 293978);