Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454681 |
| 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 | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
912 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)(9,10,13,18,14,20,15,19)(11,21,22,32,24,30,25,16)(12,26,28,34,29,33,27,17)(23,37,36,40,39,31,38,35);(1,3,4,7)(2,5,6,8)(9,11)(10,16)(12,23)(13,25)(14,24)(15,22)(17,31)(18,30)(19,21)(20,32)(26,40)(27,36)(28,38)(29,39)(33,35)(34,37);(3,7)(5,8)(11,22)(12,27)(13,15)(16,30)(17,33)(18,19)(21,32)(24,25)(26,34)(28,29)(31,37)(36,38);(9,12)(10,17)(11,23)(13,28)(14,29)(15,27)(16,31)(18,26)(19,33)(20,34)(21,35)(22,36)(24,39)(25,38)(30,40)(32,37)Polycyclic presentation:
2^9|a^2=fg,b^2=gh,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,e^b=ei,g^b=gi,h^b=hi,d^c=de,e^c=ei,g^c=gi,h^c=hi,e^d=ei,f^d=fi,g^d=gi,h^d=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 381040);