Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1380 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,4)(5,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,18)(19,27)(20,26)(21,24)(22,25)(23,28)(29,32)(30,31);(1,2)(3,4)(17,19,22,30,21,29,20,28)(18,23,26,32,24,31,25,27);(5,6,8,7)(10,13)(14,16)(18,24)(23,31)(25,26)(27,32);(2,4)(5,7,8,6)(9,11,12,15)(10,14,13,16)(19,28)(20,22)(23,27)(25,26)(29,30)(31,32)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=g,d^2=fgi,e^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,h^a=hi,d^b=dei,e^d=ei,h^d=hi140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 264061);