Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32941 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#121 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#386 |
| prime radical | #60359 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
4145 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)(5,6)(7,10,14,20)(8,13)(9,12)(11,15,19,16)(17,18);(5,7)(6,10,12,20)(8,16)(9,14)(11,18,19,13)(15,17);(5,7)(6,11)(8,16)(9,14)(10,18)(12,19)(13,20)(15,17);(2,4)Polycyclic presentation:
2^9|a^2=h,b^2=eh,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,e^b=ei,f^b=fi,h^b=hi,e^c=ei,f^c=fi,h^c=hi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72261238, #72261397, #72278601, … |
| maximal normal subgroup of | #72261238, #72261397, #72278601, … |
| maximal quotient of | #72262054 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299884);