Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60753 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27×C4 |
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 | #60797 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1902 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)(9,10,14,19,15,12,24,18)(11,16,23,30,21,20,33,29)(13,17,26,32,27,25,37,31)(22,28,36,39,35,34,40,38);(1,3,4,8)(2,5,6,7)(9,11)(10,16)(12,20)(13,22)(14,23)(15,21)(17,28)(18,29)(19,30)(24,33)(25,34)(26,36)(27,35)(31,38)(32,39)(37,40);(3,8)(5,7)(9,12)(10,15)(11,20)(13,17)(14,18)(16,21)(19,24)(22,28)(23,29)(25,27)(26,32)(30,33)(31,37)(34,35)(36,39)(38,40);(2,6)(5,7)(11,21)(16,20)(22,35)(23,33)(28,34)(29,30)(36,40)(38,39);(9,13,14,26,15,27,24,37)(10,17,19,32,12,25,18,31)(11,22,23,36,21,35,33,40)(16,28,30,39,20,34,29,38)Polycyclic presentation:
2^9|a^2=h,b^2=f,e^2=h,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=cf,d^b=di,e^c=ei,g^c=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72313983, #72315607, #72315619, … |
| maximal normal subgroup of | #72313983, #72315607, #72315619, … |
| maximal quotient of | #72343171 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6285054);