Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23995 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #535 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2487 |
| prime radical | C24 | C22×Q8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
750 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,4,3)(5,6,12,16,11,17,10,30)(7,18,22,31,21,33,20,13)(8,23,27,36,26,14,25,32)(9,28,19,34,29,35,24,15);(1,3,4,2)(5,7,10,20,11,21,12,22)(6,13,16,18,17,31,30,33)(8,24,25,29,26,19,27,9)(14,28,32,34,23,35,36,15);(5,8,11,26)(6,14)(7,19,21,24)(9,22,29,20)(10,25,12,27)(13,28)(15,33)(16,32)(17,23)(18,34)(30,36)(31,35);(5,9,10,24,11,29,12,19)(6,15,16,28,17,34,30,35)(7,8,20,25,21,26,22,27)(13,14,18,32,31,23,33,36)Polycyclic presentation:
2^9|a^2=h,b^2=gh,c^2=g,d^2=e,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,f^b=fi,d^c=di,f^c=fi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 277224);