Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500695 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #35513 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2478 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,6)(3,7,8,5)(9,10,12,14)(11,13,15,16)(19,26)(23,30)(27,32)(29,31);(1,3,4,8)(2,5,6,7)(9,11,12,15)(10,13,14,16)(17,18)(19,23)(20,22)(21,24)(25,28)(26,30)(27,29)(31,32);(10,14)(13,16)(18,22)(20,28)(23,29)(24,25)(27,32)(30,31);(2,6)(3,8)(10,14)(13,16)(17,19)(18,23)(20,27)(21,26)(22,29)(24,30)(25,31)(28,32);(18,24)(19,26)(22,25)(27,32)Polycyclic presentation:
2^9|a^2=fg,b^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=dfgi,c^b=ch,d^b=df,e^b=ei,h^b=hi,h^c=hi,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72319879 |
| maximal normal subgroup of | #72319879 |
| maximal quotient of | #72319951, #72319974 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7413162);