Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23848 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
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 | C24×D4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3316 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(9,10,12,11)(13,14,16,19)(15,17,20,18);(5,6)(7,8)(14,17)(15,20)(18,19);(2,4)(6,8)(9,11,12,10);(14,17)(15,20)(18,19)Polycyclic presentation:
2^9|a^2=h,c^2=hi,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,e^b=ei,e^d=ei104 conjugacy classes — view the full character table.
| maximal subgroup of | #72237125, #72237131, #72237141, … |
| maximal normal subgroup of | #72237125, #72237131, #72237141, … |
| maximal quotient of | #72249910, #72250314, #72250319, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 280859);