Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24986 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #555 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2571 |
| prime radical | C24 | 2−1+4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
652 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,4,3)(5,6)(7,17,20,13)(8,21)(9,12)(10,15)(11,16)(14,23)(18,29,25,26)(19,33,35,36)(22,27)(24,30)(28,32,31,34);(6,12,16,15)(7,18,20,25)(8,22)(13,17)(14,21,30,27)(19,34,35,32)(23,24)(26,29);(5,7,9,18,11,20,10,25)(6,13,12,26,16,17,15,29)(8,19,23,32,22,35,24,34)(14,28,27,33,30,31,21,36);(1,3,4,2)(5,8,10,24,11,22,9,23)(6,14,12,27,16,30,15,21)(7,19,25,34,20,35,18,32)(13,28,26,33,17,31,29,36)Polycyclic presentation:
2^9|a^2=h,b^2=eg,c^2=ef,d^2=fgh,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,d^b=di,g^c=gi,h^c=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 286911);