Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #382477 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2310 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14336 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2179 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)(3,8)(4,16,19,32)(5,10)(6,22)(7,11)(9,20,28,21)(12,26,18,31)(13,24)(14,30)(15,25)(17,23,29,27);(1,3)(2,8)(4,17,19,29)(5,13)(6,14)(7,15)(9,26,28,31)(10,24)(11,25)(12,21,18,20)(16,27,32,23)(22,30);(1,4)(2,9,11,28)(3,12,15,18)(5,20)(6,21)(7,19)(8,23)(10,16,22,32)(13,29,14,17)(24,26)(25,27)(30,31);(1,3)(2,8)(4,18)(5,13)(6,14)(7,15)(9,27)(10,24)(11,25)(12,19)(16,31)(17,20)(21,29)(22,30)(23,28)(26,32)Polycyclic presentation:
2^9|a^2=e,b^2=e,c^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,g^a=gi,h^a=hi,c^b=cg,h^b=hi,d^c=de,e^c=ei,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 289673);