Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61643 |
| 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 | #60749 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#408 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1938 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)(9,10,14,20,15,21,29,19)(11,16,17,12,23,25,32,26)(13,18,28,35,27,33,39,34)(22,30,31,24,36,38,40,37);(1,3)(2,5)(4,8)(6,7)(9,11,14,17,15,23,29,32)(10,16,20,12,21,25,19,26)(13,22,28,31,27,36,39,40)(18,30,35,24,33,38,34,37);(9,12,14,25,15,26,29,16)(10,17,20,23,21,32,19,11)(13,24,28,38,27,37,39,30)(18,31,35,36,33,40,34,22);(3,8)(5,7)(9,13)(10,18)(11,22)(12,24)(14,28)(15,27)(16,30)(17,31)(19,34)(20,35)(21,33)(23,36)(25,38)(26,37)(29,39)(32,40);(2,6)(5,7)(13,27)(18,33)(22,36)(24,37)(28,39)(30,38)(31,40)(34,35)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=h,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ci,d^b=df,g^c=gi,e^d=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72314467, #72315198, #72315586, … |
| maximal normal subgroup of | #72314467, #72315198, #72315586, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6301875);