Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573451 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2081 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2)(3,4)(5,6,9,10)(7,11,8,12)(13,14,18,22)(15,19,26,34)(16,20,29,37)(17,21,30,38)(23,31,25,33)(24,32,40,43)(27,35,28,36)(39,42,41,44);(6,10)(11,12)(13,15)(14,19)(16,24)(17,23)(18,26)(20,32)(21,31)(22,34)(25,30)(27,41)(28,39)(29,40)(33,38)(35,44)(36,42)(37,43);(2,4)(15,23)(16,27)(17,30)(19,31)(20,35)(21,38)(24,40)(25,26)(28,29)(32,43)(33,34)(36,37);(5,7,9,8)(6,11,10,12)(13,16)(14,20)(15,24)(17,28)(18,29)(19,32)(21,36)(22,37)(23,39)(25,41)(26,40)(27,30)(31,42)(33,44)(34,43)(35,38);(5,8,9,7)(6,12,10,11)(15,23)(16,28)(17,30)(19,31)(20,36)(21,38)(25,26)(27,29)(33,34)(35,37)(39,41)(42,44)Polycyclic presentation:
2^9|a^2=fi,d^2=f,e^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,h^b=hi,d^c=dh,h^c=hi,e^d=eh,h^d=hi,h^e=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6485303);