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 | C2×#3307 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2024 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2)(3,4)(5,6)(7,11)(8,10)(9,12)(13,14)(15,23)(16,20)(17,21)(18,22)(19,26)(24,34)(25,39)(27,32)(28,35)(29,38)(30,36)(31,37)(33,41)(40,44)(42,43);(5,7,9,8)(6,10,12,11)(13,15,17,26)(14,19,21,23)(16,25,30,41)(18,24,31,27)(20,33,36,39)(22,32,37,34)(28,42,29,40)(35,44,38,43);(2,4)(5,7,9,8)(6,11,12,10)(15,24)(16,28)(18,31)(19,32)(20,35)(22,37)(23,34)(25,41)(26,27)(29,30)(33,39)(36,38);(5,8,9,7)(6,10,12,11)(13,16)(14,20)(15,25)(17,30)(18,29)(19,33)(21,36)(22,38)(23,39)(24,40)(26,41)(27,42)(28,31)(32,43)(34,44)(35,37);(14,21)(15,24)(16,29)(18,31)(19,34)(20,35)(23,32)(26,27)(28,30)(33,39)(36,38)(40,42)Polycyclic presentation:
2^9|b^2=f,c^2=fi,d^2=fi,h^2=i,b^a=bf,c^a=cg,e^a=ei,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, 6484338);