Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #62307 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C22×C4×D4 |
| prime radical | #60450 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#408 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1935 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10,16,21,14,20,15,33)(11,22,26,39,24,17,25,36)(12,27,29,35,28,40,23,18)(13,19,32,34,30,37,31,38);(1,3)(2,5)(4,8)(6,7)(9,11,16,26,14,24,15,25)(10,17,21,36,20,22,33,39)(12,13,29,32,28,30,23,31)(18,34,27,37,35,38,40,19);(9,12,15,23,14,28,16,29)(10,18,21,27,20,35,33,40)(11,13,25,31,24,30,26,32)(17,34,36,37,22,38,39,19);(3,8)(5,7)(9,13)(10,19)(11,23)(12,26)(14,30)(15,31)(16,32)(17,35)(18,22)(20,37)(21,34)(24,29)(25,28)(27,39)(33,38)(36,40);(2,6)(5,7)(10,20)(12,28)(13,30)(17,22)(21,33)(23,29)(31,32)(36,39)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=g,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,d^b=df,d^c=di,e^c=ei,e^d=ei98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6296437);