Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #456658 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2688 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
703 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,4,5)(3,6,7,8)(9,10)(11,21)(12,26)(13,18)(14,19)(15,20)(16,23)(17,28)(22,37)(24,33)(25,32)(27,31)(29,35)(30,34)(36,38)(39,40);(9,11,15,25)(10,16)(12,27)(13,23,14,24)(17,31,35,37)(18,32)(19,21)(20,33)(22,30)(26,36,34,40)(28,39)(29,38);(2,5)(6,8)(9,12,15,30)(10,17,20,35)(11,22,25,27)(13,28,14,29)(16,31,33,37)(18,34,19,26)(21,36,32,40)(23,38,24,39);(1,3,4,7)(2,6,5,8)(10,18)(12,28)(13,14)(16,32)(19,20)(21,33)(22,38)(23,24)(26,34)(27,39)(29,30)(36,40)Polycyclic presentation:
2^9|a^2=ef,b^2=egh,c^2=i,d^2=ef,e^2=i,f^2=i,h^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,g^b=gi,h^b=hi,d^c=de,e^c=ei,f^c=fi,e^d=ei,f^d=fi,h^d=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 380375);