Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667836 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #31391 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#124 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1360 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,17)(12,16)(13,15)(14,19)(18,24)(20,22)(21,23)(25,26,29,32)(27,35,38,31)(28,34,30,33)(36,39,37,40);(9,11,13,19)(10,14,15,17)(12,20,21,18)(16,24,23,22)(25,27)(26,31)(28,37)(29,38)(30,36)(32,35)(33,40)(34,39);(2,5)(6,8)(26,32)(27,36)(28,30)(31,39)(35,40)(37,38);(1,3,4,7)(2,6,5,8)(10,15)(14,17)(16,23)(22,24)(25,28,29,30)(26,33,32,34)(27,37,38,36)(31,40,35,39);(9,12,13,21)(10,16,15,23)(11,18,19,20)(14,22,17,24)(25,28)(26,34)(27,38)(29,30)(31,35)(32,33)Polycyclic presentation:
2^9|a^2=i,b^2=fi,d^2=g,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei,h^c=hi,e^d=ei,h^e=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72313377, #72313531, #72313904 |
| maximal normal subgroup of | #72313377, #72313531, #72313904 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6611545);