Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417674 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2668 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | C2×#41 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
863 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,21)(12,26)(13,19)(14,18)(15,20)(16,24)(17,29)(22,37)(23,32)(25,33)(27,40)(28,36)(30,35)(31,39)(34,38);(1,3,5,8)(2,6,4,7)(9,11)(10,16)(12,27)(13,23)(14,24)(15,25)(17,34)(18,33)(19,21)(20,32)(22,28)(26,31)(29,39)(30,38)(35,37)(36,40);(9,12,15,30)(10,17,20,36)(11,22,25,39)(13,29,14,28)(16,31,32,37)(18,26,19,35)(21,34,33,40)(23,27,24,38);(1,4,5,2)(3,6,8,7)(10,18)(11,23)(13,14)(16,32)(17,26)(19,20)(22,27)(24,25)(28,29)(31,37)(35,36)(38,39)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,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,g^d=gi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 337749);