Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2192 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #447 |
| commutator | C2×C8 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1786 |
| prime radical | C2×#578 | C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C4×#131 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
803 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,3,5,4,6,7,8)(9,10,15,20,16,12,14,19)(11,21,25,37,26,38,24,35)(13,28,31,27,32,36,30,18)(17,29,34,40,22,39,33,23);(9,11)(10,17)(12,22)(13,29)(14,24)(15,25)(16,26)(18,35)(19,33)(20,34)(21,28)(23,30)(27,37)(31,40)(32,39)(36,38);(9,12,16,10)(11,22,26,17)(13,27,32,18)(14,19,15,20)(21,23,38,40)(24,33,25,34)(28,30,36,31)(29,37,39,35)Polycyclic presentation:
2^9|a^2=f,c^2=i,d^2=gi,e^2=i,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,d^b=dg,g^b=gi,e^c=ei,f^c=fi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36717);