Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3057 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C22×D4 |
| commutator | C2×C4 | C23×C8 |
| Frattini, cosocle | C4×C8 | C24 |
| socle | C22 | C2×C8×D4 |
| prime radical | C2×#135 | C8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ #2162 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
441 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,12)(3,13,4,9)(5,11,6,10)(8,15,16,14)(17,18,21,25,22,19,26,28,20,23,29,31,30,27,24,32);(1,3,7,4)(2,8,12,16)(5,14,6,15)(9,10,13,11)(18,23)(19,27)(25,31)(28,32);(1,4,7,3)(2,9,12,13)(5,14,6,15)(8,10,16,11)(17,19,21,28,22,23,26,31,20,27,29,32,30,18,24,25);(1,5,7,6)(2,10,12,11)(3,14,4,15)(8,13,16,9)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=f,d^2=g,e^2=g,f^2=h,h^2=i,b^a=be,c^a=ci,d^a=dg,e^a=eg,c^b=ci,e^b=eg,d^c=dg128 conjugacy classes — view the full character table.
| maximal subgroup of | #72320765, #72320835, #72320867, … |
| maximal normal subgroup of | #72320765, #72320835, #72320867, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 390957);