Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23886 |
| 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 | #2366 |
| prime radical | #23421 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1389 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,3)(5,6,11,16,12,18,10,17)(7,19,22,13,23,33,31,34)(8,24,26,14,27,35,32,21)(9,20,28,15,30,36,29,25);(1,2,4,3)(5,7,12,23)(6,13,18,34)(8,25,27,15)(9,21,30,14)(10,31,11,22)(16,33,17,19)(20,32,36,26)(24,29,35,28);(5,8)(6,14)(7,20)(9,19)(10,32)(11,26)(12,27)(13,28)(15,22)(16,35)(17,24)(18,21)(23,36)(25,31)(29,34)(30,33);(1,3,4,2)(5,9,10,29,12,30,11,28)(6,15,17,20,18,25,16,36)(7,21,31,35,23,14,22,24)(8,19,32,34,27,33,26,13)Polycyclic presentation:
2^9|a^2=h,b^2=ef,d^2=efh,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,e^c=ei,f^c=fi,g^c=gi,e^d=ei,f^d=fi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 285313);