Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×D8 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C2×C4 | C23×C8 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C22×#135 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C4×C22⋊C8 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1198 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2)(3,4)(5,6,7,9,8,10,11,12)(14,19)(15,17)(16,23)(20,24)(21,25)(27,28);(13,14,18,21)(15,19,17,25)(16,24,23,27)(20,26,28,22);(2,4)(13,15,18,17)(14,19,21,25)(16,22,23,26)(20,27,28,24);(13,16,18,23)(14,20,21,28)(15,22,17,26)(19,27,25,24)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=h,d^2=h,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eh,d^b=de,e^b=eh,e^d=eh140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 127327);