Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23848 |
| 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×C4×M16 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
999 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)(13,14,18,21)(15,22,24,27)(16,20,17,25)(19,26,28,23);(13,15)(14,19)(16,23)(17,26)(18,24)(20,27)(21,28)(22,25);(2,4)(6,7)(11,12)(13,16,18,17)(14,20,21,25)(15,23,24,26)(19,27,28,22);(5,7,9,12,8,6,10,11)Polycyclic presentation:
2^9|a^2=g,c^2=h,d^2=gh,e^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,e^b=eh,d^c=di104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 116407);