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) = C2×C43 |
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 | |
1212 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,4,3)(9,10)(11,14)(12,13)(15,16)(17,18,21,23,22,26,20,25)(19,24,28,30,29,32,27,31);(5,6,7,8)(18,23)(19,27)(20,21)(25,26)(28,29)(30,31);(1,2,4,3)(10,13)(14,16);(1,3,4,2)(9,11,12,15)(10,14,13,16)(17,19,21,28,22,29,20,27)(18,24,23,30,26,32,25,31)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=eg,d^2=fg,e^2=i,g^2=i,b^a=be,c^a=cf,d^b=de,e^b=ei,g^b=gi140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 264086);