Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417802 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2669 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1023 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(2,5)(6,8)(9,10)(11,20)(12,24)(13,27)(14,19)(15,22)(16,25)(17,23)(18,30)(21,31)(26,36)(28,35)(29,32)(33,39)(34,40)(37,38);(1,2)(3,6)(4,5)(7,8)(9,11,13,22,14,23,30,29)(10,15,18,20,19,32,27,17)(12,21,25,38,26,39,28,40)(16,33,24,37,35,31,36,34);(1,3,4,7)(2,6,5,8)(9,12,14,26)(10,16,19,35)(11,21,23,39)(13,28,30,25)(15,31,32,33)(17,34,20,37)(18,36,27,24)(22,40,29,38);(2,5)(6,8)(9,11)(10,17)(12,21)(13,29)(14,23)(15,27)(16,34)(18,32)(19,20)(22,30)(24,31)(25,40)(26,39)(28,38)(33,36)(35,37)Polycyclic presentation:
2^9|b^2=f,c^2=ef,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,g^b=gi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi,g^d=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 333059);