Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23848 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C22×C4⋊C4 |
| prime radical | C24 | C2×C4⋊C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
686 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6,9,12)(7,11,14,19)(8,10,16,13)(15,17,20,18)(21,22,25,29,26,28,24,33)(23,30,32,27,31,35,34,36);(6,10)(8,16)(11,17)(12,13)(15,20)(18,19)(21,23,24,34,26,31,25,32)(22,27,33,30,28,36,29,35);(1,2,3,4)(5,7)(6,11)(8,15)(9,14)(10,17)(12,19)(13,18)(16,20);(6,12)(7,14)(10,13)(15,20)(22,28)(23,31)(29,33)(32,34)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=f,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eh,d^b=di,e^b=eh,d^c=dhi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 110902);