Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6342800 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×C22⋊C4 |
| commutator | C23 | C24×C4 |
| Frattini, cosocle | C24 | C25 |
| prime radical | C2×#3499 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4⋊Q8 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
4104 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,16)(12,23,26,17)(13,18)(14,19)(15,20)(21,31,24,29)(22,30)(25,36,38,32)(27,33)(28,34)(35,40,37,39);(1,3)(2,5)(4,8)(6,7)(9,11,15,13)(10,16,20,18)(12,21,26,24)(14,27,28,22)(17,29,23,31)(19,33,34,30)(25,37,38,35)(32,40,36,39);(9,12)(10,17)(11,21)(13,24)(14,25)(15,26)(16,29)(18,31)(19,32)(20,23)(22,35)(27,37)(28,38)(30,39)(33,40)(34,36);(3,8)(5,7)(9,13,15,11)(10,18,20,16)(12,24,26,21)(14,22,28,27)(17,31,23,29)(19,30,34,33)(25,35,38,37)(32,39,36,40);(2,6)(5,7)(9,14,15,28)(10,19,20,34)(11,22,13,27)(12,25,26,38)(16,30,18,33)(17,32,23,36)(21,35,24,37)(29,39,31,40)Polycyclic presentation:
2^9|a^2=h,b^2=i,d^2=i,e^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,d^b=df,e^b=ei,h^c=hi,e^d=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72302902, #72302906, #72303141, … |
| maximal normal subgroup of | #72302902, #72302906, #72303141, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6293178);