Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470402 |
| 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 |
| prime radical | C2×#540 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2030 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)(13,14,19,24)(15,25,28,41)(16,21,17,22)(18,32,33,23)(20,34,36,27)(26,40,30,43)(29,39,31,37)(35,44,38,42);(7,8)(10,12)(13,15,18,27,19,28,33,34)(14,20,23,25,24,36,32,41)(16,26,29,42,17,30,31,44)(21,35,37,40,22,38,39,43);(1,2,3,4)(5,7,9,8)(6,10,11,12)(13,16)(14,21)(15,26)(17,19)(18,29)(20,35)(22,24)(23,37)(25,40)(27,42)(28,30)(31,33)(32,39)(34,44)(36,38)(41,43);(5,8,9,7)(6,10,11,12)(13,17,19,16)(14,22,24,21)(15,26,28,30)(18,31,33,29)(20,35,36,38)(23,39,32,37)(25,40,41,43)(27,42,34,44);(6,11)(10,12)(16,17)(21,22)(26,30)(29,31)(35,38)(37,39)(40,43)(42,44)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=g,d^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dhi,d^c=di,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6390580);