Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573615 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3269 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2167 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)(7,10)(8,11)(9,12)(13,14)(15,20)(16,27)(17,23)(18,31)(19,22)(21,28)(24,39)(25,36)(26,34)(29,38)(30,35)(32,37)(33,42)(40,44)(41,43);(1,2,3,4)(5,7)(6,9)(8,12)(10,11)(13,15,19,25)(14,20,22,36)(16,24,29,40)(17,30,18,26)(21,33,37,43)(23,35,31,34)(27,39,38,44)(28,42,32,41);(7,12)(9,10)(13,16)(14,21)(15,24)(17,28)(18,32)(19,29)(20,33)(22,37)(23,38)(25,40)(26,41)(27,31)(30,42)(34,39)(35,44)(36,43);(5,7)(6,10)(8,12)(9,11)(14,22)(15,25)(21,37)(23,31)(24,40)(26,30)(27,38)(41,42);(7,12)(9,10)(13,17,19,18)(14,23,22,31)(15,26,25,30)(16,28,29,32)(20,34,36,35)(21,38,37,27)(24,41,40,42)(33,39,43,44)Polycyclic presentation:
2^9|b^2=fhi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,g^a=gi,c^b=ch,d^b=di,e^b=ehi,d^c=dh,g^c=gi,e^d=eh77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6504201);