Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61071 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C23×C22⋊C4 |
| prime radical | #62812 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4749 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 20 points:
(2,4)(5,6,9,13,10,8,17,12)(7,11,16,19,15,14,20,18);(1,2)(3,4)(5,7)(6,11)(8,14)(9,16)(10,15)(12,18)(13,19)(17,20);(5,8)(6,10)(7,14)(9,12)(11,15)(13,17)(16,18)(19,20);(2,4)(5,7)(6,11)(8,14)(9,16)(10,15)(12,18)(13,19)(17,20);(7,15)(11,14)(16,20)(18,19)Polycyclic presentation:
2^9|a^2=h,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,d^b=df,e^b=ei,g^c=gi,e^d=ei110 conjugacy classes — view the full character table.
| maximal subgroup of | #72286003, #72311454, #72312083, … |
| maximal normal subgroup of | #72286003, #72311454, #72312083, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6286308);