Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30412 |
| 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 | C2×#2408 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3078 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)(9,10,11,12)(13,14,16,18,17,20,15,19);(5,6)(7,8)(9,10)(11,12);(2,4)(6,8)(14,18)(15,16)(19,20);(2,4)(10,12)(14,19)(15,16)(18,20)Polycyclic presentation:
2^9|a^2=h,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=defi,c^b=cg,d^b=de,f^c=fi,h^c=hi,f^d=fi,h^d=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72239750, #72239763, #72239765, … |
| maximal normal subgroup of | #72239750, #72239763, #72239765, … |
| maximal quotient of | #72257521, #72257586 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 290294);