Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454041 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2668 |
| 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 | |
928 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,5,7)(3,6,4,8)(9,10)(11,21)(12,26)(13,19)(14,18)(15,20)(16,23)(17,28)(22,38)(24,33)(25,32)(27,37)(29,36)(30,35)(31,40)(34,39);(1,3,5,4)(2,6,7,8)(9,11)(10,16)(12,27)(13,24)(14,23)(15,25)(17,34)(18,32)(19,21)(20,33)(22,28)(26,38)(29,40)(30,39)(31,35)(36,37);(2,7)(6,8)(9,12,15,30)(10,17,20,36)(11,22,25,40)(13,28,14,29)(16,31,33,38)(18,26,19,35)(21,37,32,34)(23,27,24,39);(1,4,5,3)(2,8,7,6)(12,28)(13,14)(17,26)(18,19)(21,32)(22,27)(23,24)(29,30)(31,37)(34,38)(35,36)(39,40)Polycyclic presentation:
2^9|a^2=ef,b^2=ef,c^2=i,d^2=ef,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,h^b=hi,d^c=de,e^c=ei,f^c=fi,g^c=gi,e^d=ei,f^d=fi,g^d=gi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 366966);