Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535815 |
| 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 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2549 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)(18,32)(19,23)(20,26)(25,39)(27,34)(28,36)(29,37)(30,35)(31,38)(33,42)(40,43)(41,44);(5,6)(7,10)(8,9)(11,12)(13,15,18,26,19,27,17,30)(14,20,22,24,23,35,32,34)(16,25,28,42,29,41,31,40)(21,33,38,39,37,43,36,44);(1,2,3,4)(6,9)(10,12)(13,16,19,29)(14,21,23,37)(15,25,27,41)(17,31,18,28)(20,33,35,43)(22,38,32,36)(24,39,34,44)(26,42,30,40);(5,7)(6,10)(8,11)(9,12)(13,17,19,18)(14,22,23,32)(15,26,27,30)(16,28,29,31)(20,34,35,24)(21,36,37,38)(25,40,41,42)(33,39,43,44);(7,11)(10,12)(16,29)(21,37)(25,41)(28,31)(33,43)(36,38)(39,44)(40,42)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=eh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6465311);