Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #213798 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2308 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #35660 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1774 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,5)(4,6)(7,8)(9,10)(11,16)(12,17)(13,18)(14,19)(15,20)(21,29)(22,30)(23,31)(24,32)(25,33)(26,34)(27,35)(28,36)(37,39)(38,40);(1,3)(2,5)(4,7)(6,8)(9,11)(10,16,19,32)(12,25,13,22)(14,24)(15,23)(17,33)(18,30)(20,31,36,40)(21,26,37,27)(28,38)(29,35)(34,39);(2,6)(5,8)(9,12)(10,17)(11,21)(13,14)(15,27)(16,29)(18,19)(20,35)(22,23)(24,37)(25,38)(26,28)(30,31)(32,39)(33,40)(34,36);(3,7)(5,8)(9,13,14,12)(10,18,19,17)(11,22,24,25)(15,26,28,27)(16,30,32,33)(20,34,36,35)(21,23,37,38)(29,31,39,40)Polycyclic presentation:
2^9|b^2=h,d^2=i,g^2=i,b^a=be,c^a=cf,h^a=hi,c^b=cg,d^b=def,e^b=ei,d^c=de,g^c=gi,h^c=hi,g^d=gi,h^d=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72262952, #72263055, #72263057, … |
| maximal normal subgroup of | #72262952, #72263055, #72263057, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 298915);