Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437537 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3521 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3158 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,25)(16,30)(17,24)(18,22)(19,23)(20,29)(21,33)(26,41)(27,37)(28,36)(31,44)(32,39)(34,40)(35,42)(38,43);(2,4)(13,15,18,28)(14,20,22,37)(16,31,33,42)(17,27,19,29)(21,38,30,41)(23,36,24,25)(26,32,43,34)(35,39,44,40);(6,9)(10,12)(13,16,18,33)(14,21,22,30)(15,26,28,43)(17,34,19,32)(20,35,37,44)(23,40,24,39)(25,38,36,41)(27,42,29,31);(15,27)(17,19)(20,36)(23,24)(25,37)(26,42)(28,29)(31,43)(32,34)(35,41)(38,44)(39,40);(5,7,8,11)(6,10,9,12)(14,22)(15,28)(21,30)(23,24)(26,43)(27,29)(31,42)(39,40)Polycyclic presentation:
2^9|b^2=i,c^2=i,e^2=gi,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,h^a=hi,c^b=ch,d^b=dhi,e^b=ei,f^b=fi,h^b=hi,f^c=fi,h^c=hi,f^d=fi,h^d=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6361232);