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 | #392 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2046 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(5,6)(7,12)(8,10)(9,11)(13,14)(15,25)(16,30)(17,22)(18,23)(19,24)(20,28)(21,31)(26,36)(27,39)(29,42)(32,40)(33,43)(34,41)(35,38)(37,44);(5,7,9,8)(6,10,11,12)(13,15)(14,20)(16,27)(17,33)(18,28)(19,35)(21,39)(22,36)(23,25)(24,42)(26,32)(29,34)(30,37)(31,44)(38,41)(40,43);(1,2,3,4)(13,16)(14,21)(15,26)(17,19)(18,31)(20,36)(22,41)(23,30)(24,40)(25,43)(27,35)(28,33)(29,44)(32,34)(37,38)(39,42);(5,8,9,7)(6,10,11,12)(16,17)(19,34)(21,40)(22,30)(24,41)(26,44)(27,33)(29,35)(31,32)(36,37)(38,42)(39,43);(13,17,18,32)(14,22,23,40)(15,27,28,44)(16,19,31,34)(20,37,25,39)(21,41,30,24)(26,35,33,29)(36,38,43,42)Polycyclic presentation:
2^9|b^2=fi,c^2=gi,d^2=fi,e^2=i,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, 6491565);