Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573643 |
| 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 | #61056 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2025 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)(3,6)(4,5)(7,8)(9,10)(11,23)(12,27)(13,20)(14,21)(15,18)(16,22)(17,25)(19,26)(24,36)(28,40)(29,33)(30,37)(31,34)(32,38)(35,39);(2,5)(6,8)(9,11)(10,17)(12,16)(13,28)(14,25)(15,26)(18,27)(19,22)(20,35)(21,23)(24,31)(29,32)(30,39)(33,34)(36,38)(37,40);(1,3,4,7)(2,6,5,8)(10,18)(11,12)(15,16)(19,27)(20,34)(21,22)(24,28)(25,26)(29,39)(31,32)(35,40)(37,38);(9,12,14,26)(10,19,21,27)(11,16,25,15)(13,29,30,24)(17,22,23,18)(20,36,37,33)(28,32,39,31)(34,35,38,40);(9,13)(10,20)(11,24)(12,28)(14,30)(15,31)(16,32)(17,33)(18,34)(19,35)(21,37)(22,38)(23,36)(25,29)(26,39)(27,40)Polycyclic presentation:
2^9|c^2=fi,d^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,g^a=gi,c^b=ch,e^b=eh,h^b=hi,d^c=dh,g^c=gi,h^c=hi,e^d=eh,h^d=hi,h^e=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6482162);