Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #484095 |
| 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×#509 | C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #1451 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
456 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,4,7,5,3,6,8)(9,10,14,18)(11,21)(12,25,26,19)(13,17,28,36)(15,30,29,20)(16,32)(22,34)(23,37)(24,33)(27,38)(31,39)(35,40);(9,11)(10,16)(12,22)(13,27)(14,24)(15,31)(17,33)(18,34)(19,37)(20,38)(21,36)(23,29)(25,39)(26,32)(28,40)(30,35);(1,3,4,8,5,2,6,7)(9,12,15,13,14,26,29,28)(10,17,20,25,18,36,30,19)(11,22,31,27,24,32,23,40)(16,33,38,39,34,21,35,37);(2,3)(7,8)(10,18)(16,34)(17,36)(19,25)(20,30)(21,33)(35,38)(37,39)Polycyclic presentation:
2^9|a^2=f,c^2=fgi,e^2=g,f^2=h,g^2=i,b^a=be,c^a=cg,d^a=dhi,e^a=eg,g^a=gi,e^b=egi,f^b=fi,g^b=gi,d^c=dh,e^c=ei62 conjugacy classes — view the full character table.
| maximal subgroup of | #72216752, #72216779, #72216810, … |
| maximal normal subgroup of | #72216752, #72216779, #72216810, … |
| maximal quotient of | C2×#484105, #72318666, #72318674, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 391301);