Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470028 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3497 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4013 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)(4,7)(5,8)(9,10,14,17)(11,19,21,35)(12,16,25,33)(13,26,29,18)(15,27,31,22)(20,38,23,37)(24,36,28,34)(30,40,32,39);(3,4)(6,7)(9,11)(10,15)(12,23)(13,27)(14,21)(16,32)(17,31)(18,35)(19,26)(20,25)(22,29)(24,39)(28,40)(30,33)(34,38)(36,37);(1,3,5,4)(2,6,8,7)(9,12,14,25)(10,16,17,33)(11,20,21,23)(13,28,29,24)(15,30,31,32)(18,36,26,34)(19,37,35,38)(22,40,27,39);(1,4,5,3)(2,7,8,6)(10,17)(11,21)(16,33)(18,26)(20,23)(22,27)(34,36)(39,40);(2,8)(6,7)(10,17)(15,31)(16,33)(18,26)(19,35)(30,32)(34,36)(37,38)Polycyclic presentation:
2^9|a^2=i,c^2=h,d^2=hi,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fi,c^b=ch,d^b=dh,f^b=fi,g^b=gi,f^c=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6394693);