Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9103 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #386 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
2498 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)(9,10)(11,19,22,15)(12,23)(13,26)(14,17)(16,24)(18,29)(20,36,38,40)(21,31,27,34)(25,35)(28,33)(30,37,32,39);(2,4)(5,6,7,8)(9,11)(10,15)(12,20)(13,27)(14,22)(16,32)(17,19)(18,34)(21,29)(23,40)(24,39)(25,38)(26,31)(28,37)(30,33)(35,36);(9,12,14,25)(10,16,17,33)(11,20,22,38)(13,28,29,24)(15,30,19,32)(18,35,26,23)(21,39,27,37)(31,36,34,40)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=i,e^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,f^a=fi,c^b=cf,e^b=ei,g^b=gi,e^c=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72314501, #72314509, #72314536, … |
| maximal normal subgroup of | #72314501, #72314509, #72314536, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30757);