Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682648 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #60458 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1916 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)(3,7)(4,6)(5,8)(9,10)(11,20)(12,26)(13,19)(14,18)(15,24)(16,29)(17,28)(21,34)(22,33)(23,25)(27,37)(30,36)(31,35)(32,38)(39,40);(1,3,4,8)(2,5,6,7)(9,11)(10,15)(12,27)(13,25)(14,24)(16,34)(17,23)(18,20)(19,33)(21,31)(22,28)(26,32)(29,40)(30,38)(35,39)(36,37);(9,12,28,31,14,30,13,29)(10,16,19,36,18,35,17,26)(11,21,22,38,24,40,25,27)(15,32,33,34,20,37,23,39);(2,6)(5,7)(9,13,14,28)(10,17,18,19)(11,22,24,25)(12,29,30,31)(15,33,20,23)(16,26,35,36)(21,38,40,27)(32,34,37,39);(3,8)(5,7)(9,10)(11,23)(12,16)(13,17)(14,18)(15,22)(19,28)(20,25)(21,39)(24,33)(26,29)(27,37)(30,35)(31,36)(32,38)(34,40)Polycyclic presentation:
2^9|b^2=fi,c^2=gi,d^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,h^b=hi,h^c=hi,h^e=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6590468);