Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23886 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C2×D42 |
| prime radical | #23889 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2060 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)(11,20)(12,25)(13,18)(14,30)(15,19)(16,23)(17,28)(21,34)(22,24)(26,37)(27,36)(29,35)(31,33)(32,39)(38,40);(2,5)(6,8)(9,11)(10,16)(12,26)(13,23)(14,24)(15,31)(17,34)(18,20)(19,22)(21,29)(25,32)(27,39)(28,38)(30,33)(35,40)(36,37);(9,12,13,27)(10,17,18,35)(11,21,23,38)(14,29,15,28)(16,32,20,37)(19,36,30,25)(22,34,33,40)(24,39,31,26);(9,10)(11,22)(12,17)(13,18)(14,19)(15,30)(16,31)(20,24)(21,34)(23,33)(25,28)(26,32)(27,35)(29,36)(37,39)(38,40);(1,3,4,7)(2,6,5,8)Polycyclic presentation:
2^9|c^2=i,e^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,h^b=hi,g^c=gi,h^c=hi,h^d=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6346945);