Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30437 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2668 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14357 |
| prime radical | #60387 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1929 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,11,8,12,7,10)(3,13,18,27,17,28,16,9)(4,5,20,24,21,25,26,23)(14,30,29,15,31,22,19,32);(1,3)(2,9)(4,19)(5,22)(6,18)(7,16)(8,17)(10,28)(11,13)(12,27)(14,20)(15,23)(21,29)(24,32)(25,30)(26,31);(1,4)(2,5)(3,14)(6,20)(7,26)(8,21)(9,15)(10,23)(11,24)(12,25)(13,22)(16,19)(17,31)(18,29)(27,32)(28,30);(1,5)(2,4)(3,15)(6,24)(7,23)(8,25)(9,14)(10,26)(11,20)(12,21)(13,29)(16,30)(17,32)(18,22)(19,28)(27,31)Polycyclic presentation:
2^9|a^2=h,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,g^c=gi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 290367);