Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2180 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #442 |
| commutator | C8 | C23×C8 |
| Frattini, cosocle | C4×C8 | C24 |
| socle | C22 | C2×#442 |
| prime radical | C2×#508 | C4 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×#322 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
498 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,4,7,5,3,6,8)(9,10,14,19,15,20,12,17)(11,21,22,38,25,39,24,35)(13,27,30,36,29,34,26,18)(16,31,32,23,33,28,37,40);(9,11)(10,16)(12,22)(13,28)(14,24)(15,25)(17,32)(18,35)(19,37)(20,33)(21,34)(23,30)(26,40)(27,39)(29,31)(36,38);(1,3,4,8,5,2,6,7)(10,17,20,19)(13,29)(16,32,33,37)(18,27,36,34)(21,35,39,38)(23,40)(26,30)(28,31);(9,12,15,14)(10,17,20,19)(11,22,25,24)(13,26,29,30)(16,32,33,37)(18,34,36,27)(21,38,39,35)(23,28,40,31)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=cg,e^a=eg,g^a=gi,e^b=egi,f^b=fi,g^b=gi,e^c=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72216219, #72216441, #72216444, … |
| maximal normal subgroup of | #72216219, #72216441, #72216444, … |
| maximal quotient of | C4×#12067, C2×#483589, #72227152, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 390785);