Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C8 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C4×C8 | C24 |
| commutator, socle | C22 | C22×C4×C8 |
| prime radical | C2×#41 | C2×C8 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
566 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,4,8,5,6,11,13)(3,7,10,14,9,12,15,16)(17,18,20,25,22,26,23,21)(19,27,29,30,28,24,31,32);(2,6)(7,12)(8,13)(14,16)(17,19,21,30,23,31,26,27,22,28,25,32,20,29,18,24);(1,3)(2,7)(4,10)(5,9)(6,12)(8,14)(11,15)(13,16)(19,28)(24,27)(29,31)(30,32);(3,9)(7,12)(10,15)(14,16)Polycyclic presentation:
2^9|a^2=e,b^2=f,e^2=g,f^2=h,h^2=i,b^a=bg,c^b=ci,d^c=dgi200 conjugacy classes — view the full character table.
| maximal subgroup of | #72231889, #72231892, #72231893, … |
| maximal normal subgroup of | #72231889, #72231892, #72231893, … |
| maximal quotient of | #72323178 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 387149);