Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #480315 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×#410 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×#41 | C24 |
| socle | C2 | C2×#1799 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×M16 ◁ C2×#568 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
1117 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)(3,12)(4,16)(5,10,19,8)(6,11)(7,14)(9,18)(13,26)(15,27,29,25)(17,23,31,24)(20,30,22,32)(21,28);(1,3,4,14,6,13,18,28)(2,7,9,12,11,21,16,26)(5,15,17,30,19,29,31,32)(8,20,23,25,10,22,24,27);(2,8)(5,19)(7,20)(9,23)(10,11)(12,25)(15,29)(16,24)(17,31)(21,22)(26,27)(30,32);(3,13)(7,21)(12,26)(14,28)(15,29)(20,22)(25,27)(30,32)Polycyclic presentation:
2^9|a^2=g,b^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fh,h^a=hi,d^b=di,f^c=fi,g^c=ghi,g^f=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72236900, #72237535, #72237550, … |
| maximal normal subgroup of | #72236900, #72237535, #72237550, … |
| maximal quotient of | #72311867, C2×#493737, #72337510 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400933);