Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #365053 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2363 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #13984 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1115 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)(3,12,16,23)(4,17)(5,9,13,10)(7,21,24,15)(8,20)(14,30)(18,28)(19,27)(22,32)(25,31)(26,29);(1,3,13,21,6,16,5,15)(2,7,9,23,11,24,10,12)(4,18,19,32,20,14,29,31)(8,25,27,30,17,22,26,28);(1,4)(2,8)(3,14)(5,19)(6,20)(7,22)(9,26)(10,27)(11,17)(12,28)(13,29)(15,31)(16,18)(21,32)(23,30)(24,25);(2,9,11,10)(7,23,24,12)(8,26,17,27)(22,30,25,28)Polycyclic presentation:
2^9|a^2=h,b^2=eh,d^2=f,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,g^b=gi,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72318354, #72318940, #72318981, … |
| maximal normal subgroup of | #72318354, #72318940, #72318981, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 273193);