Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23994 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #535 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2580 |
| prime radical | C24 | C2×#41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
824 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,11,16,12,17,10,27)(7,18,23,29,22,33,21,13)(8,14,24,31,25,15,9,26)(19,34,36,28,35,32,20,30);(1,2,3,4)(5,7)(6,13,17,29)(8,19)(9,20)(10,21)(11,23)(12,22)(14,30,15,28)(16,18,27,33)(24,36)(25,35)(26,32,31,34);(2,4)(5,8,11,24,12,25,10,9)(6,14,16,31,17,15,27,26)(7,19,23,36,22,35,21,20)(13,28,18,32,29,30,33,34);(5,9,10,25,12,24,11,8)(6,15,16,26,17,14,27,31)(7,20,21,35,22,36,23,19)(13,28,18,32,29,30,33,34)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=h,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=fi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 282262);