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 | #2619 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2034 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)(15,25,30,41)(16,22,32,21)(17,23,18,34)(20,28,36,29)(26,42,27,40)(31,38,33,37)(35,43,39,44);(7,8)(10,12)(13,15,18,29,19,30,17,28)(14,20,23,25,24,36,34,41)(16,26,31,43,32,27,33,44)(21,35,38,40,22,39,37,42);(1,2,3,4)(5,7,9,8)(6,10,11,12)(13,16,19,32)(14,21,24,22)(15,26,30,27)(17,33,18,31)(20,35,36,39)(23,38,34,37)(25,40,41,42)(28,44,29,43);(5,8,9,7)(6,10,11,12)(13,16)(14,22)(15,27)(17,33)(18,31)(19,32)(20,35)(21,24)(23,37)(25,40)(26,30)(28,43)(29,44)(34,38)(36,39)(41,42);(6,11)(10,12)(13,17,19,18)(14,23,24,34)(15,28,30,29)(16,31,32,33)(20,25,36,41)(21,37,22,38)(26,43,27,44)(35,42,39,40)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=g,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=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, 6390582);