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)(3,13,16,27)(4,17,18,10)(5,19,21,11)(6,9)(7,12)(8,15,23,32)(14,29,31,28)(20,24,30,25)(22,26);(1,3)(2,8)(4,15)(5,20)(6,16)(7,14)(9,25)(10,13)(11,27)(12,24)(17,29)(18,30)(19,28)(21,32)(22,31)(23,26);(2,9)(3,14)(8,23)(10,11)(12,26)(13,28)(15,30)(16,31)(17,19)(20,32)(24,25)(27,29);(1,4,6,5)(2,10,9,11)(3,15,16,20)(7,18,22,21)(8,13,25,27)(12,17,26,19)(14,30,31,32)(23,28,24,29)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,e^b=eh,f^b=fgh,f^d=fi,f^e=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72283617, #72283632, #72283678, … |
| maximal normal subgroup of | #72283617, #72283632, #72283678, … |
| maximal quotient of | #72284212 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 389751);