Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30428 |
| 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 | D4×C22⋊C4 |
| prime radical | #60453 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1399 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)(7,18)(8,23,25,34)(9,15,27,36)(10,28)(11,16)(12,17)(13,21)(14,24,20,30)(19,26,35,31)(22,32)(29,33);(1,2)(3,4)(5,7)(6,13)(8,19)(9,23)(10,29)(11,21)(12,22)(14,31)(15,24)(16,32)(17,33)(18,28)(20,26)(25,35)(27,34)(30,36);(2,4)(5,8)(6,14)(7,19)(9,18)(10,24)(11,30)(12,25)(13,31)(15,29)(16,34)(17,20)(21,36)(22,35)(23,28)(26,33)(27,32);(5,9,10,26,12,27,11,31)(6,15,16,19,17,36,28,35)(7,20,21,23,22,14,29,34)(8,18,24,33,25,32,30,13)Polycyclic presentation:
2^9|a^2=h,d^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,c^b=cg,d^b=de,e^b=ei,f^c=fi,h^c=hi,f^d=fi,h^d=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 290732);