Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3290 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2358 |
| prime radical | C2×#585 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2331 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,4)(5,6)(7,15,17,19)(8,11)(9,13)(10,14)(12,16,20,18);(1,2)(3,4)(6,11,14,13)(12,15,20,19);(5,7)(6,12)(8,16)(9,18)(10,17)(11,15)(13,19)(14,20);(2,4)(5,6)(7,12)(8,11)(9,13)(10,14)(15,16)(17,20)(18,19)Polycyclic presentation:
2^9|a^2=g,b^2=h,e^2=i,f^2=i,b^a=be,c^a=cf,e^a=ei,h^a=hi,d^b=dei,f^c=fi,g^c=gi,e^d=ei,h^d=hi116 conjugacy classes — view the full character table.
| maximal subgroup of | #72226293, #72241172, #72241274, … |
| maximal normal subgroup of | #72226293, #72241172, #72241274, … |
| maximal quotient of | #72209991, #1042, #72238300, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 263265);