Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573511 |
| 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×#3285 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2058 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14,18,19)(15,20,26,23)(16,21,30,28)(17,22,33,34)(24,35,27,36)(25,31,38,39)(29,37,43,32)(40,41,42,44);(6,9)(10,12)(13,15,18,26)(14,20,19,23)(16,27,30,24)(17,31,33,39)(21,36,28,35)(22,38,34,25)(29,41,43,44)(32,40,37,42);(2,4)(5,7,8,11)(6,10,9,12)(14,19)(15,23)(16,28)(20,26)(21,30)(22,34)(25,39)(29,32)(31,38)(35,36)(37,43)(41,44);(13,16,18,30)(14,21,19,28)(15,24,26,27)(17,32,33,37)(20,35,23,36)(22,29,34,43)(25,41,38,44)(31,42,39,40);(13,17,18,33)(14,22,19,34)(15,25,26,38)(16,29,30,43)(20,31,23,39)(21,37,28,32)(24,40,27,42)(35,41,36,44)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=f,d^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,e^b=ehi,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6482068);