Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535823 |
| 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 | #39357 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2524 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(13,14)(15,24)(16,21)(17,22)(18,33)(19,23)(20,26)(25,41)(27,35)(28,44)(29,38)(30,39)(31,36)(32,40)(34,43)(37,42);(5,6)(7,10)(8,9)(11,12)(13,15,18,26,19,27,17,31)(14,20,22,24,23,36,33,35)(16,28,29,42,30,25,32,43)(21,37,40,44,39,34,38,41);(1,2,3,4)(6,9)(10,12)(13,16,19,30)(14,21,23,39)(15,25,27,28)(17,32,18,29)(20,34,36,37)(22,40,33,38)(24,41,35,44)(26,43,31,42);(5,7)(6,10)(8,11)(9,12)(13,17,19,18)(14,22,23,33)(15,26,27,31)(16,29,30,32)(20,35,36,24)(21,38,39,40)(25,42,28,43)(34,41,37,44);(7,11)(10,12)(16,30)(21,39)(25,28)(29,32)(34,37)(38,40)(41,44)(42,43)Polycyclic presentation:
2^9|b^2=f,c^2=gi,d^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=di,d^c=di,e^c=ei,e^d=ehi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6465319);