Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #483839 |
| 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 | C2×#508 | C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #1450 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
446 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,5,3,4,6,8,7)(9,10,13,20,15,21,29,18)(11,22,25,17,12,26,23,37)(14,30,28,19,32,38,39,33)(16,31,34,24,36,40,35,27);(2,6)(3,7)(9,11)(10,16)(12,15)(13,23)(14,31)(17,19)(18,34)(20,35)(21,36)(22,38)(24,39)(25,29)(26,30)(27,28)(32,40)(33,37);(1,3,5,6,4,7,8,2)(9,12,15,11)(10,17)(13,23,29,25)(14,24,32,27)(16,33)(18,22)(19,36)(20,26)(21,37)(28,40,39,31)(30,34)(35,38);(9,13,15,29)(10,18,21,20)(11,23,12,25)(14,28,32,39)(16,34,36,35)(17,26,37,22)(19,30,33,38)(24,31,27,40)Polycyclic presentation:
2^9|a^2=f,c^2=fg,d^2=i,e^2=g,f^2=h,g^2=i,b^a=be,c^a=cgh,d^a=di,e^a=eg,g^a=gi,c^b=ch,e^b=egi,f^b=fi,g^b=gi,d^c=di,e^c=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72216701, #72216702, #72216730, … |
| maximal normal subgroup of | #72216701, #72216702, #72216730, … |
| maximal quotient of | C2×#483844, #72318455, #72318708, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 391040);