Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #455564 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2622 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
911 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,4,6)(3,5,7,8)(9,10)(11,15)(12,23)(13,16)(14,19)(17,24)(18,26)(20,28)(21,27)(22,30)(25,32)(29,35)(31,37)(33,36)(34,38)(39,40);(1,3)(2,5)(4,7)(6,8)(9,11)(10,15)(12,20)(13,21)(14,22)(16,27)(17,28)(18,29)(19,30)(23,36)(24,33)(25,34)(26,35)(31,38)(32,39)(37,40);(3,7)(5,8)(10,16)(12,24)(13,26)(15,27)(17,32)(18,19)(20,33)(21,35)(23,31)(28,39)(29,30)(36,38);(9,12,13,25,14,24,26,37)(10,17,18,32,19,23,16,31)(11,20,21,34,22,33,35,40)(15,28,29,39,30,36,27,38)Polycyclic presentation:
2^9|a^2=g,d^2=f,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,h^b=hi,d^c=de,f^c=fi,e^d=ei,h^d=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 369509);