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) = C23×C42 |
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 | |
1286 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,4)(5,6,7,8)(13,14,18,19,17,20,16,22)(15,21,25,26,24,27,23,28);(9,10,11,12)(14,19)(15,23)(16,18)(20,22)(24,25)(26,28);(1,2,3,4)(14,20)(19,22)(21,27)(26,28);(13,15,18,25,17,24,16,23)(14,21,19,26,20,27,22,28)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=fi,d^2=ei,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, 263562);