Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #508304 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #13304 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
616 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)(3,6)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16);(2,5)(6,8)(17,18,19,24,27,31,32,29,21,23,28,30,20,25,26,22);(1,3)(2,6)(4,7)(5,8)(10,13)(14,16)(18,22)(19,26)(20,27)(23,29)(24,25)(28,32)(30,31);(3,7)(6,8)(9,11,12,15)(10,14,13,16)(18,23)(22,29)(24,30)(25,31)Polycyclic presentation:
2^9|b^2=g,d^2=f,g^2=hi,h^2=i,b^a=be,c^a=cf,c^b=cg,d^b=di,d^c=de,g^c=gh,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72343806, #72343840, #72343842, … |
| maximal normal subgroup of | #72343806, #72343840, #72343842, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 416187);