Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23924 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C22×#41 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×#431 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
967 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)(5,6,9,11,8,7,10,12)(13,14)(15,24)(16,21)(17,22)(18,27)(19,23)(20,25)(26,28);(6,7)(11,12)(13,15,19,16)(14,20,23,28)(17,25,18,26)(21,27,24,22);(2,4)(13,16,19,15)(14,21,23,24)(17,25,18,26)(20,27,28,22);(5,7,9,12,8,6,10,11)(13,17,19,18)(14,22,23,27)(15,25,16,26)(20,21,28,24)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=h,d^2=gh,e^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,d^b=di,e^b=eh,d^c=dh92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 116403);