Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454664 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2622 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
786 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(3,7)(5,8)(9,10,14,18)(11,20)(12,16)(13,17,29,19)(15,28)(21,33,39,37)(22,32)(23,31)(24,30,38,40)(25,35)(26,34)(27,36);(1,2,4,6)(3,8,7,5)(9,11,14,23)(10,15,18,32)(12,24,27,38)(13,28,29,22)(16,33,36,37)(17,31,19,20)(21,25,39,26)(30,34,40,35);(1,3,4,7)(2,5,6,8)(9,12,13,26,14,27,29,25)(10,16,19,35,18,36,17,34)(11,21,22,24,23,39,28,38)(15,30,31,33,32,40,20,37);(10,17)(12,25)(13,29)(15,31)(18,19)(20,32)(21,38)(22,28)(24,39)(26,27)(30,40)(34,35)Polycyclic presentation:
2^9|a^2=efg,b^2=gh,c^2=g,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,f^b=fi,g^b=gi,h^b=hi,d^c=de,f^c=fi,e^d=ei,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, 381943);