Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12087 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #442 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C4×C8 | C24 |
| socle | C22 | C2×#442 |
| prime radical | C2×#508 | C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×#322 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
452 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)(9,10,14,17,15,20,12,18)(11,21,23,38,22,39,25,36)(13,27,30,37,29,35,26,19)(16,31,32,24,33,28,34,40);(2,6)(3,7)(9,11)(10,16)(12,23)(13,28)(14,25)(15,22)(17,34)(18,32)(19,36)(20,33)(21,35)(24,30)(26,40)(27,39)(29,31)(37,38);(1,3,5,6,4,7,8,2)(10,17,20,18)(11,22)(13,29)(16,32,33,34)(19,35,37,27)(21,36,39,38)(23,25)(26,30);(9,12,15,14)(10,18,20,17)(11,23,22,25)(13,26,29,30)(16,32,33,34)(19,35,37,27)(21,38,39,36)(24,28,40,31)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,e^a=eg,g^a=gi,c^b=ch,e^b=egi,f^b=fi,g^b=gi,e^c=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72216707, #72216726, #72216752, … |
| maximal normal subgroup of | #72216707, #72216726, #72216752, … |
| maximal quotient of | C2×#483836, #72318524, #72318692, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 391032);