Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #481619 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#99 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C4⋊C4 | C24 |
| prime radical | #35692 | C2 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×#518 ◁ G | |
| LECS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1112 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,12,7,13,16,32)(3,10,17,8,15,26,20,27)(4,19,18,31,21,11,23,29)(6,14,22,25,24,9,28,30);(2,8)(4,20)(5,22)(9,19)(10,25)(11,14)(12,31)(13,27)(16,28)(17,21)(26,30)(29,32);(2,9)(3,15)(5,16)(8,19)(10,31)(11,27)(12,25)(13,14)(18,23)(22,28)(26,29)(30,32);(1,3,6,18)(2,10,14,31)(4,16,20,28)(5,17,22,21)(7,15,24,23)(8,25,11,12)(9,29,13,26)(19,32,27,30)Polycyclic presentation:
2^9|a^2=f,d^2=h,e^2=h,f^2=i,b^a=be,c^a=cgh,e^a=eg,g^a=gi,e^b=eh,f^b=fgh,d^c=di,f^c=fi,f^e=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72318712 |
| maximal normal subgroup of | #72318712 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 389493);