Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32932 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C23 | C23×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4692 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,4)(5,6)(7,8)(9,10)(11,20)(12,17)(13,18)(14,29)(15,19)(16,23)(21,35)(22,31)(24,36)(25,39)(26,33)(27,34)(28,40)(30,38)(32,37);(9,11,14,23,15,24,13,22)(10,16,18,20,19,31,29,36)(12,25,28,37,26,21,27,38)(17,32,34,39,33,30,40,35);(2,4)(9,12)(10,17)(11,21)(13,27)(14,28)(15,26)(16,30)(18,34)(19,33)(20,35)(22,37)(23,38)(24,25)(29,40)(31,32)(36,39);(6,8)(12,26)(17,33)(21,25)(27,28)(30,32)(34,40)(35,39)(37,38);(9,13,15,14)(10,18,19,29)(11,22,24,23)(12,27,26,28)(16,20,31,36)(17,34,33,40)(21,37,25,38)(30,35,32,39)Polycyclic presentation:
2^9|b^2=f,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,c^b=ci,d^c=di104 conjugacy classes — view the full character table.
| maximal subgroup of | #72315560, #72315565, #72315572, … |
| maximal normal subgroup of | #72315560, #72315565, #72315572, … |
| maximal quotient of | #72315743 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6536202);