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 ◁ C4×#95 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1478 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2,4,3)(5,6,8,7)(13,14,16,20)(15,22,23,18)(17,19,25,21)(24,27,28,26);(2,3)(9,10,11,12)(13,15)(14,18)(16,23)(17,24)(19,26)(20,22)(21,27)(25,28);(5,6,8,7)(14,19)(17,25)(18,26)(20,21)(22,27)(24,28);(1,3,4,2)(5,7,8,6)(15,23)(18,22)(24,28)(26,27)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=eg,d^2=gi,f^2=i,b^a=be,c^a=cf,f^a=fi,d^b=de,f^c=fi140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 263891);