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 | C2×#3267 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2527 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)(20,28)(25,40)(26,35)(27,41)(29,44)(30,37)(31,39)(32,38)(34,42)(36,43);(5,6)(7,10)(8,9)(11,12)(13,15,17,26,19,27,18,28)(14,20,33,41,23,35,22,24)(16,29,32,42,31,25,30,43)(21,36,37,40,39,34,38,44);(1,2,3,4)(6,9)(10,12)(13,16,19,31)(14,21,23,39)(15,25,27,29)(17,32,18,30)(20,34,35,36)(22,38,33,37)(24,40,41,44)(26,43,28,42);(5,7)(6,10)(8,11)(9,12)(13,17,19,18)(14,22,23,33)(15,26,27,28)(16,30,31,32)(20,24,35,41)(21,37,39,38)(25,42,29,43)(34,44,36,40);(7,11)(10,12)(15,27)(20,35)(24,41)(25,29)(26,28)(34,36)(40,44)(42,43)Polycyclic presentation:
2^9|b^2=fi,c^2=gi,d^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=ei,d^c=di,e^d=ehi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6465483);