Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #481601 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#99 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C22⋊C4 | C24 |
| prime radical | #35190 | C2 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
994 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,11,13,16)(4,9)(5,10)(6,8)(7,12,15,14)(17,18)(19,23,21,24)(20,25)(22,26)(27,30,29,31)(28,32);(1,3)(2,7)(4,12)(5,14)(6,13)(8,15)(9,11)(10,16)(18,23)(19,21)(20,28)(24,26)(25,31)(30,32);(2,8)(7,15)(9,10)(11,16)(17,19,22,21)(18,24,26,23)(20,27,28,29)(25,31,32,30);(1,4,6,5)(2,9,8,10)(3,12,13,14)(7,11,15,16)(17,20,22,28)(18,25,26,32)(19,27,21,29)(23,30,24,31)Polycyclic presentation:
2^9|a^2=f,c^2=i,d^2=h,e^2=h,b^a=be,c^a=cgh,e^a=eg,g^a=gi,c^b=ci,d^b=di,e^b=eh,f^b=fgh,f^c=fi,f^e=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72316687, #72316690, #72316706, … |
| maximal normal subgroup of | #72316687, #72316690, #72316706, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 389449);