Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60388 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3018 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4610 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 20 points:
(1,2,4,3)(6,11)(8,9)(12,19)(13,17)(14,16)(18,20);(5,6)(7,12)(8,11)(9,17)(10,13)(14,18)(15,20)(16,19);(5,7)(6,12)(8,14)(9,16)(10,15)(11,18)(13,20)(17,19);(1,3,4,2)(5,7)(6,12)(8,16)(9,14)(10,15)(11,19)(13,20)(17,18);(5,8,10,9)(6,11,13,17)(7,14,15,16)(12,18,20,19)Polycyclic presentation:
2^9|a^2=h,d^2=h,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,f^b=fi,g^b=gi,f^c=fi,e^d=ei116 conjugacy classes — view the full character table.
| maximal subgroup of | #72227324, #72227326, C4×#60388, … |
| maximal normal subgroup of | #72227326, C4×#60388, #72310464, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6279913);