Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454664 |
| 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×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
896 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,4,6)(3,5,7,8)(9,10)(11,21,25,31)(12,26)(13,16)(14,19)(15,20)(17,24,33,22)(18,28)(23,39,40,32)(27,34,38,37)(29,36)(30,35);(1,3,4,7)(2,5,6,8)(10,16)(11,22)(13,14)(18,26)(19,20)(21,31)(23,38)(24,25)(27,40)(28,29)(34,37)(35,36);(3,7)(5,8)(9,11)(10,17,20,33)(12,27,30,38)(13,22)(14,24)(15,25)(16,31,19,21)(18,34)(23,28,40,29)(26,39)(32,35)(36,37);(9,12,15,30)(10,18,20,36)(11,23,25,40)(13,29,14,28)(16,26,19,35)(17,32,33,39)(21,37,31,34)(22,38,24,27)Polycyclic presentation:
2^9|a^2=eg,b^2=gh,c^2=ef,d^2=i,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,h^b=hi,d^c=de,g^c=gi,h^c=hi,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, 381255);