Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30434 |
| 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 | #60359 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2014 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,10,7,11,5,20)(3,12,14,8,15,25,21,23)(4,16,18,9,19,28,22,27)(13,30,32,29,31,24,17,26);(1,3)(2,8)(4,17)(5,21)(6,14)(7,15)(9,26)(10,25)(11,23)(12,20)(13,18)(16,24)(19,32)(22,31)(27,29)(28,30);(8,23)(12,25)(24,30)(26,29);(1,4)(2,9)(3,13)(5,22)(6,18)(7,19)(8,24)(10,28)(11,27)(12,29)(14,32)(15,31)(16,20)(17,21)(23,30)(25,26)Polycyclic presentation:
2^9|a^2=h,e^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,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72213409, #72216047, #72216154, … |
| maximal normal subgroup of | #72213409, #72216047, #72216154, … |
| maximal quotient of | #72250505, C2×#383149, #72256104, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 290345);