Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24292 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
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 | C22×#135 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1340 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6,12,16,11,17,10,18)(7,19,23,32,22,35,21,13)(8,24,27,36,26,14,25,34)(9,28,20,33,30,31,29,15);(5,7)(6,13,17,32)(8,9)(10,21)(11,22)(12,23)(14,33,24,15)(16,19,18,35)(20,27)(25,29)(26,30)(28,34,31,36);(1,2,3,4)(5,8,11,26)(6,14,17,24)(7,20,22,29)(9,23,30,21)(10,25,12,27)(13,31,32,28)(15,35,33,19)(16,34,18,36);(5,9,10,29,11,30,12,20)(6,15,16,28,17,33,18,31)(7,8,21,25,22,26,23,27)(13,14,19,34,32,24,35,36)Polycyclic presentation:
2^9|a^2=h,b^2=eh,c^2=f,d^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,g^b=gi,d^c=di,g^c=gi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 272823);