Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #185255 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#379 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #62814 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
2349 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,7,10,21)(3,12)(4,15)(8,22,24,31)(9,19,11,16)(13,28)(14,27)(17,30)(20,23,32,25)(26,29);(1,2)(3,8)(4,16)(5,9)(6,11)(7,17)(10,18)(12,25)(13,22)(14,23)(15,21)(19,30)(20,26)(24,27)(28,31)(29,32);(4,15)(7,19)(13,29)(16,21)(17,30)(20,31)(22,32)(26,28);(1,3,5,14)(2,8,9,23)(4,13,15,29)(6,12,18,27)(7,20,19,31)(10,24,11,25)(16,22,21,32)(17,26,30,28)Polycyclic presentation:
2^9|a^2=g,d^2=h,e^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,e^b=ei,f^b=fh,g^b=ghi,e^c=eh74 conjugacy classes — view the full character table.
| maximal subgroup of | #72213377, #72213800, #72213843, … |
| maximal normal subgroup of | #72213377, #72213800, #72213843, … |
| maximal quotient of | C2×#185257, #72240256, #72240270 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 92453);