Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30484 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2358 |
| prime radical | C2×#3018 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2012 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(6,12,14,13)(8,15,18,19);(5,6)(7,15)(8,16)(9,13)(10,12)(11,14)(17,19)(18,20);(2,4)(5,7)(6,8)(9,16)(10,20)(11,17)(12,15)(13,19)(14,18);(2,4)(5,8)(6,7)(9,19)(10,15)(11,18)(12,20)(13,16)(14,17)Polycyclic presentation:
2^9|a^2=h,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,g^c=gi,e^d=ei,h^d=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72267112 |
| maximal normal subgroup of | #72267112 |
| maximal quotient of | #72209991, #72238302, #72238304, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 290243);