Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470237 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C24 | C25 |
| prime radical | #62811 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
4294 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2)(3,5)(4,6)(7,8)(9,10,15,19)(11,16,23,30)(12,17,26,32)(13,27,14,18)(20,33,36,28)(21,37,22,29)(24,38,25,31)(34,39,35,40);(1,3)(2,5)(4,7)(6,8)(9,11)(10,16)(12,20)(13,21)(14,22)(15,23)(17,28)(18,29)(19,30)(24,34)(25,35)(26,36)(27,37)(31,39)(32,33)(38,40);(3,7)(5,8)(9,12)(10,17)(11,20)(13,24)(14,25)(15,26)(16,28)(18,31)(19,32)(21,34)(22,35)(23,36)(27,38)(29,39)(30,33)(37,40);(3,7)(5,8)(9,13,15,14)(10,18,19,27)(11,21,23,22)(12,24,26,25)(16,29,30,37)(17,31,32,38)(20,34,36,35)(28,39,33,40);(2,6)(5,8)(9,14,15,13)(10,18,19,27)(11,22,23,21)(12,25,26,24)(16,29,30,37)(17,31,32,38)(20,35,36,34)(28,39,33,40)Polycyclic presentation:
2^9|a^2=i,d^2=i,e^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dh,g^b=gi,f^c=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72279295, #72279464, #72303498, … |
| maximal normal subgroup of | #72279295, #72279464, #72303498, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6377438);