Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6469983 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2050 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)(15,24)(16,21)(17,22)(18,31)(19,23)(20,26)(25,39)(27,34)(28,36)(29,37)(30,38)(32,35)(33,42)(40,44)(41,43);(7,8)(10,12)(13,15)(14,20)(16,25)(17,26)(18,32)(19,27)(21,33)(22,34)(23,35)(24,31)(28,40)(29,41)(30,42)(36,43)(37,39)(38,44);(1,2,3,4)(5,7,9,8)(6,10,11,12)(13,16,19,28)(14,21,23,36)(15,25,27,40)(17,30,18,29)(20,33,35,43)(22,38,31,37)(24,39,34,44)(26,42,32,41);(5,8,9,7)(6,10,11,12)(16,28)(21,36)(25,40)(29,30)(33,43)(37,38)(39,44)(41,42);(6,11)(10,12)(13,17,19,18)(14,22,23,31)(15,26,27,32)(16,29,28,30)(20,34,35,24)(21,37,36,38)(25,41,40,42)(33,39,43,44)Polycyclic presentation:
2^9|c^2=gi,d^2=h,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,f^b=fi,d^c=di,e^c=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6390430);