Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2021 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #120 |
| commutator | C24 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C23 | #385 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
2546 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)(3,4)(5,6)(7,8)(9,10)(11,15,20,28)(12,22,23,16)(13,25)(14,17)(18,26)(19,30)(21,35,34,29)(24,31,37,36)(27,33)(32,38)(39,40);(2,4)(9,11,13,21,14,20,26,34)(10,15,18,29,17,28,25,35)(12,19,24,32,23,33,37,40)(16,30,31,39,22,27,36,38);(6,8)(9,12)(10,16)(11,19)(13,24)(14,23)(15,27)(17,22)(18,31)(20,33)(21,32)(25,36)(26,37)(28,30)(29,38)(34,40)(35,39)Polycyclic presentation:
2^9|a^2=g,b^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,f^a=fi,h^a=hi,c^b=cf,d^b=di,g^b=gi,d^c=di,g^c=gi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72314504, #72314522, #72314526, … |
| maximal normal subgroup of | #72314504, #72314522, #72314526, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30787);