Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417682 |
| 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 | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1039 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)(9,10,15,20)(11,21,24,36)(12,25,28,33)(13,17,14,19)(16,30,32,23)(18,27,35,29)(22,31,38,40)(26,37,39,34);(1,3)(2,6)(4,7)(5,8)(9,11)(10,16)(12,26)(13,30)(14,23)(15,24)(17,21)(18,34)(19,36)(20,32)(22,27)(25,40)(28,39)(29,38)(31,33)(35,37);(1,4,5,2)(3,7,8,6)(10,17)(13,14)(16,21)(18,33)(19,20)(23,30)(25,35)(26,39)(27,29)(31,37)(32,36)(34,40);(3,8)(6,7)(9,12)(10,18)(11,22)(13,29)(14,27)(15,28)(16,31)(17,33)(19,25)(20,35)(21,37)(23,39)(24,38)(26,30)(32,40)(34,36)Polycyclic presentation:
2^9|a^2=ef,c^2=efi,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,d^b=de,e^b=ei,f^b=fi,e^c=ei,f^c=fi,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, 332295);