Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6574015 |
| 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×#469 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#533 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1823 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)(5,6,9,8)(7,10,12,11)(13,14)(15,24)(16,21)(17,22)(18,32)(19,23)(20,28)(25,37)(26,38)(27,34)(29,35)(30,36)(31,42)(33,40)(39,43)(41,44);(6,8)(10,11)(13,15,18,28,19,26,17,27)(14,20,22,24,23,34,32,38)(16,25,31,40,29,41,30,39)(21,33,36,37,35,43,42,44);(2,4)(5,6,9,8)(7,11,12,10)(13,16)(14,21)(15,25)(17,30)(18,31)(19,29)(20,33)(22,36)(23,35)(24,37)(26,41)(27,39)(28,40)(32,42)(34,43)(38,44);(5,7,9,12)(6,10,8,11)(15,26)(16,29)(20,34)(21,35)(24,38)(27,28)(30,31)(36,42);(5,8,9,6)(7,10,12,11)(13,17,19,18)(14,22,23,32)(15,27,26,28)(16,30,29,31)(20,24,34,38)(21,36,35,42)(25,39,41,40)(33,37,43,44)Polycyclic presentation:
2^9|a^2=h,b^2=fh,c^2=h,d^2=h,e^2=hi,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=di,e^b=eh,d^c=dhi,e^d=eh80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6534549);