Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #482362 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
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 | #35050 | C2 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ C23×C22⋊C4 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1421 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)(3,13,17,29)(4,18,19,10)(5,20,22,11)(6,9)(7,12)(8,16,24,32)(14,25,31,30)(15,27,21,26)(23,28);(1,3)(2,8)(4,16)(5,21)(6,17)(7,14)(9,27)(10,13)(11,29)(12,26)(15,19)(18,25)(20,30)(22,32)(23,31)(24,28);(2,9)(3,14)(8,24)(10,11)(12,28)(13,30)(15,16)(17,31)(18,20)(21,32)(25,29)(26,27);(1,4,6,5)(2,10,9,11)(3,15,17,32)(7,19,23,22)(8,25,27,30)(12,18,28,20)(13,24,29,26)(14,16,31,21)Polycyclic presentation:
2^9|a^2=f,d^2=h,e^2=h,b^a=be,c^a=ch,d^a=dg,e^a=eg,g^a=gi,c^b=ci,d^b=di,e^b=eh,f^b=fgh,f^d=fi,f^e=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72285245 |
| maximal normal subgroup of | #72285245 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 389763);