Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13307 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #447 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1786 |
| prime radical | #60695 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C4×#131 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
631 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,4,7,5,3,8,6)(9,10,14,20,16,21,15,33)(11,22,24,18,12,25,27,38)(13,28,30,37,32,39,31,19)(17,29,34,26,36,40,35,23);(1,3)(2,5)(4,6)(7,8)(9,11)(10,17)(12,16)(13,29)(14,24)(15,27)(18,37)(19,38)(20,34)(21,36)(22,28)(23,31)(25,39)(26,30)(32,40)(33,35);(2,3)(6,7)(9,12,16,11)(10,18,21,38)(13,26,32,23)(14,24,15,27)(17,19,36,37)(20,22,33,25)(28,34,39,35)(29,31,40,30)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,c^b=ch,d^b=dg,g^b=gi,e^c=ei,f^c=fi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36760);