Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454041 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2668 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1091 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(2,5)(6,8)(9,10,15,20)(11,21,25,31)(12,26,30,32)(13,16,14,19)(17,23,34,24)(18,28,36,29)(22,37,39,33)(27,35,38,40);(1,2,4,5)(3,6,7,8)(10,16)(13,14)(17,31)(18,32)(19,20)(21,34)(23,24)(26,36)(27,38)(28,29)(33,40)(35,37);(1,3)(2,6)(4,7)(5,8)(9,11)(10,17)(12,27)(13,24)(14,23)(15,25)(16,31)(18,35)(19,21)(20,34)(22,28)(26,37)(29,39)(30,38)(32,33)(36,40);(3,7)(6,8)(9,12)(10,18)(11,22)(13,29)(14,28)(15,30)(16,32)(17,33)(19,26)(20,36)(21,35)(23,38)(24,27)(25,39)(31,40)(34,37)Polycyclic presentation:
2^9|a^2=i,b^2=ef,e^2=i,f^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,e^b=ei,f^b=fi,h^b=hi,d^c=de,e^c=ei,f^c=fi,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, 362901);