Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437547 |
| 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×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2135 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2,4,3)(5,6)(7,10)(8,12)(9,11)(13,14)(15,25)(16,22)(17,23)(18,21)(19,24)(20,29)(26,34)(27,41)(28,39)(30,44)(31,38)(32,36)(33,40)(35,43)(37,42);(2,3)(5,7,9,8)(6,10,11,12)(13,15)(14,20)(16,30)(17,28)(18,29)(19,26)(21,25)(22,37)(23,34)(24,39)(27,31)(32,42)(33,43)(35,38)(36,44)(40,41);(1,2,4,3)(5,8,9,7)(6,10,11,12)(14,21)(15,26)(17,19)(20,34)(22,36)(25,39)(27,30)(28,29)(31,33)(35,44)(37,41)(42,43);(1,3,4,2)(13,16,18,32)(14,22,21,36)(15,27,29,43)(17,31,19,33)(20,35,25,41)(23,38,24,40)(26,30,28,42)(34,44,39,37);(13,17,18,19)(14,23,21,24)(15,28,29,26)(16,31,32,33)(20,34,25,39)(22,38,36,40)(27,42,43,30)(35,44,41,37)Polycyclic presentation:
2^9|a^2=fi,b^2=gi,c^2=fg,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,h^b=hi,e^c=ei,h^c=hi,h^d=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6354312);