Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32932 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
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 | C2×#496 | C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ #1744 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
819 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,4)(5,6,8,7,9,10,12,11)(13,14)(15,22)(16,20)(17,27)(18,21)(19,25)(23,24)(26,28);(13,15,17,25,18,26,16,24)(14,19,20,22,21,23,27,28);(1,2,3,4)(13,14)(15,23)(16,27)(17,20)(18,21)(19,26)(22,24)(25,28);(5,7,8,10,9,11,12,6)(13,16,18,17)(14,20,21,27)(15,24,26,25)(19,22,23,28)Polycyclic presentation:
2^9|a^2=g,b^2=e,c^2=f,d^2=gh,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,c^b=ch,d^c=dh104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 120166);