Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33830 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D8 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C4×C8 | C24 |
| socle | C22 | C2×C4×D8 |
| prime radical | #33733 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C4×M16 ◁ C8×M16 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
472 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,9,5,8,14,16)(3,10,12,13,11,7,15,4)(17,18,20,23)(19,22,28,30)(21,29,27,25)(24,32,31,26);(1,3)(2,7)(4,9)(5,11)(6,12)(8,10)(13,16)(14,15)(18,22)(19,26)(21,27)(23,30)(24,25)(28,32)(29,31);(1,4,6,10,5,13,14,7)(2,3,9,12,8,11,16,15)(18,23)(22,30)(24,31)(25,29);(3,11)(4,13)(7,10)(12,15)Polycyclic presentation:
2^9|a^2=f,c^2=fi,e^2=gi,f^2=h,g^2=i,b^a=be,c^a=ci,e^a=eg,g^a=gi,d^b=dh,e^b=eg,g^b=gi,d^c=dh86 conjugacy classes — view the full character table.
| maximal subgroup of | #72232027, #72234326, #72234327, … |
| maximal normal subgroup of | #72232027, #72234326, #72234327, … |
| maximal quotient of | #72234326, #72234328, #72234347, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 391314);