Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30484 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×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 | C2×#575 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1102 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,11,17,12,18,10,16)(7,19,22,13,23,31,21,32)(8,24,25,14,26,33,20,34)(9,27,28,36,29,15,30,35);(1,2)(3,4)(5,7)(6,13)(8,9)(10,22)(11,21)(12,23)(14,15)(16,31)(17,19)(18,32)(20,28)(24,35)(25,30)(26,29)(27,34)(33,36);(2,4)(5,8,10,25,12,26,11,20)(6,14,16,33,18,34,17,24)(7,9,22,30,23,29,21,28)(13,15,31,36,32,27,19,35);(5,9)(6,15)(7,20)(8,22)(10,30)(11,28)(12,29)(13,24)(14,31)(16,36)(17,35)(18,27)(19,34)(21,26)(23,25)(32,33)Polycyclic presentation:
2^9|a^2=h,c^2=efh,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,e^b=ei,h^b=hi,f^c=fi,h^c=hi,e^d=ei,f^d=fi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 296379);