Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2184 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D8 |
| commutator | C2×C8 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle, prime radical | C23 | C4×D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C4×C8 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
386 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 3 generators on 44 points:
(2,4)(5,6,8,11,9,7,10,12)(13,14,17,24,19,26,34,21,20,27,16,23,18,25,36,39)(15,28,30,41,32,43,35,42,33,38,29,40,31,44,37,22);(1,2,3,4)(5,7,8,12,9,6,10,11)(14,21)(16,34)(17,36)(18,19)(22,38)(23,25)(24,26)(27,39)(28,42)(29,35)(30,37)(31,32)(40,44)(41,43);(13,15,20,33)(14,22,27,42)(16,35,17,37)(18,32,19,31)(21,38,39,28)(23,43,24,44)(25,41,26,40)(29,36,30,34)Polycyclic presentation:
2^9|a^2=f,b^2=dfgi,c^2=i,d^2=gi,e^2=gi,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^b=dg,e^b=eg,f^b=fg,g^b=gi,h^b=hi,d^c=dg,e^c=eg,f^c=fg,g^c=gi,h^c=hi116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43884);