Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437547 |
| 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×#3516 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3104 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)(13,14,20,17)(15,25,30,27)(16,22,33,32)(18,24,19,23)(21,29,36,28)(26,38,42,41)(31,37,43,44)(34,40,35,39);(2,4)(13,15,20,30)(14,21,17,36)(16,31,33,43)(18,29,19,28)(22,38,32,41)(23,27,24,25)(26,34,42,35)(37,39,44,40);(6,9)(10,12)(13,16,20,33)(14,22,17,32)(15,26,30,42)(18,35,19,34)(21,37,36,44)(23,40,24,39)(25,38,27,41)(28,31,29,43);(13,17,20,14)(15,27,30,25)(16,32,33,22)(18,23,19,24)(21,28,36,29)(26,41,42,38)(31,44,43,37)(34,39,35,40);(5,7,8,11)(6,10,9,12)(16,33)(22,32)(26,42)(31,43)(34,35)(37,44)(38,41)(39,40)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=i,d^2=i,e^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,h^a=hi,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^c=ei,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, 6361287);