Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394358 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2410 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D8 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1429 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,5,7)(3,8,4,6)(9,10)(11,15)(12,16)(13,17)(14,18)(19,26)(20,34)(21,29)(22,28)(23,27)(24,33)(25,31)(30,36)(32,38)(35,40)(37,39);(1,3,5,4)(2,6,7,8)(9,11)(10,15)(12,23)(13,22)(14,21)(16,27)(17,28)(18,29)(19,30)(20,24)(25,37)(26,36)(31,39)(32,40)(33,34)(35,38);(1,4,5,3)(2,6,7,8)(9,12)(10,16)(11,20,21,23)(13,25)(14,24)(15,27,29,34)(17,31)(18,33)(19,32)(22,35,36,37)(26,38)(28,39,30,40);(10,17)(13,26)(15,28)(16,31)(18,19)(22,36)(25,38)(27,39)(29,30)(32,33)(34,40)(35,37)Polycyclic presentation:
2^9|a^2=e,b^2=e,c^2=ef,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,f^b=fi,g^c=gi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 305257);