Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6994139 |
| 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×#3257 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1792 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)(6,10)(8,11)(13,14)(15,23)(16,20)(17,30)(18,21)(19,26)(22,31)(24,37)(25,33)(27,36)(28,34)(29,42)(32,40)(35,41)(38,43)(39,44);(5,6,9,10)(7,8,12,11)(13,15)(14,19)(16,24)(17,25)(18,27)(20,32)(21,33)(22,36)(23,30)(26,31)(28,39)(29,38)(34,43)(35,44)(37,42)(40,41);(2,4)(5,7)(6,11)(8,10)(9,12)(13,16,18,28)(14,20,21,34)(15,24,27,39)(17,29,31,41)(19,32,33,43)(22,35,30,42)(23,37,36,44)(25,38,26,40);(6,10)(7,12)(14,21)(15,25)(17,31)(19,23)(20,34)(24,38)(26,27)(29,41)(32,37)(33,36)(39,40)(43,44);(5,8,9,11)(6,7,10,12)(14,21)(16,28)(19,33)(22,30)(23,36)(24,39)(29,41)(38,40)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=i,e^2=h,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ehi,f^a=fi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei,f^d=fi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7089377);