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 | #2361 |
| prime radical | C2×#3067 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1365 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,10,17,11,16,31,15)(7,18,22,32,21,25,20,12)(8,23,26,19,27,13,28,33)(9,24,29,35,30,34,36,14);(5,7)(6,12)(8,24)(9,23)(10,22)(11,21)(13,30)(14,28)(15,25)(16,32)(17,18)(19,29)(20,31)(26,35)(27,34)(33,36);(2,4)(5,8)(6,13)(7,14)(9,25)(10,26)(11,27)(12,29)(15,19)(16,23)(17,33)(18,30)(20,34)(21,35)(22,24)(28,31)(32,36);(2,4)(5,9,10,29,11,30,31,36)(6,14,15,34,16,35,17,24)(7,19,20,23,21,33,22,13)(8,25,26,12,27,18,28,32)Polycyclic presentation:
2^9|a^2=h,d^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,c^b=cg,d^b=de,e^b=ei,g^b=gi,g^c=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 290731);