Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3055 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C23×C43 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2046 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)(3,4)(5,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,27)(20,26)(21,24)(22,25)(23,29)(28,32)(30,31);(2,4)(17,19,22,30,21,28,20,29)(18,23,26,32,24,31,25,27);(10,13)(14,16)(18,24)(19,28)(25,26)(29,30);(9,11,12,15)(10,14,13,16)(19,28)(23,31)(27,32)(29,30)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=fi,e^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,h^a=hi,c^b=ci,d^b=di128 conjugacy classes — view the full character table.
| maximal subgroup of | #72240744, #72240777, #72240812, … |
| maximal normal subgroup of | #72240744, #72240777, #72240812, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 260986);