Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #483807 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#442 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C4×C8 | C24 |
| prime radical | C22×D16 | C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #1450 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
496 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,4,3,5,6,8,7)(9,10,13,19,14,17,30,20)(11,21,24,35,22,34,38,16)(12,25,28,18,29,15,26,32)(23,33,40,37,27,36,39,31);(10,15)(12,26)(13,30)(16,31)(17,25)(18,19)(20,32)(21,36)(23,39)(24,38)(27,40)(28,29)(33,34)(35,37);(1,3,4,6,5,7,8,2)(9,11)(10,16,17,35)(12,27)(13,24)(14,22)(15,31,25,37)(18,36,32,33)(19,21,20,34)(23,29)(26,40)(28,39)(30,38);(10,17)(11,22)(15,25)(18,32)(19,20)(23,27)(24,38)(39,40)Polycyclic presentation:
2^9|a^2=f,c^2=fgi,e^2=g,f^2=h,g^2=i,b^a=be,c^a=cgh,d^a=di,e^a=eg,g^a=gi,e^b=egi,f^b=fi,g^b=gi,d^c=di,e^c=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | #72216715, #72216720, #72216898, … |
| maximal normal subgroup of | #72216715, #72216720, #72216898, … |
| maximal quotient of | C2×#483813, #72318518, #72321367, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 391009);