Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #576 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C8 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C4 | C22×C4×C8 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C22×#41 | C2×C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×#431 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1252 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)(5,6,8,11,9,7,10,12)(14,19)(16,17)(20,26)(21,22)(24,25)(27,28);(2,4)(13,14,17,22,18,21,16,19)(15,20,25,27,23,28,24,26);(5,7,8,12,9,6,10,11)(13,15)(14,20)(16,24)(17,25)(18,23)(19,26)(21,28)(22,27);(14,21)(15,23)(19,22)(24,25)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=f,e^2=h,f^2=i,g^2=h,b^a=be,e^a=eh,g^a=gh,d^b=dh,d^c=dh176 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 89604);