Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30428 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2361 |
| prime radical | #23438 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1207 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,3,4)(5,6,12,15,11,9,10,16)(7,17,21,27,20,31,19,13)(8,22,26,32,25,14,24,30)(18,29,35,28,34,33,23,36);(5,7,10,19,11,20,12,21)(6,13,15,17,9,27,16,31)(8,23,24,34,25,35,26,18)(14,29,30,28,22,33,32,36);(2,4)(5,8,10,24,11,25,12,26)(6,14,16,32,9,22,15,30)(7,18,19,23,20,34,21,35)(13,28,31,29,27,36,17,33);(2,4)(5,9)(6,11)(7,17)(8,22)(10,15)(12,16)(13,19)(14,25)(18,33)(20,31)(21,27)(23,28)(24,30)(26,32)(29,34)(35,36)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=fh,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,c^b=cg,d^b=de,g^b=gi,g^c=gi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 296312);