Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #420420 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1385 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(2,6)(7,8)(9,10)(11,21)(12,15)(13,28,32,18)(14,19)(16,20)(17,23)(22,35,38,39)(24,34)(25,26)(27,31,36,30)(29,33,37,40);(1,2,5,6)(3,7,4,8)(9,11)(10,17)(12,26)(13,29)(14,24)(15,23)(16,25)(18,35)(19,21)(20,34)(22,30)(27,33)(28,39)(31,38)(32,37)(36,40);(1,3,5,4)(2,7,6,8)(9,12,15,10,16,19,14,20)(11,17,24,26,25,34,23,21)(13,27,31,18,32,36,30,28)(22,33,37,39,38,40,29,35);(1,4,5,3)(2,7,6,8)(9,13,14,30,16,32,15,31)(10,18,12,27,20,28,19,36)(11,22,23,29,25,38,24,37)(17,33,21,35,34,40,26,39)Polycyclic presentation:
2^9|a^2=fg,b^2=eg,c^2=ei,d^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^b=hi,h^c=hi,f^d=fi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 348342);