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 | C22×#135 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1985 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)(5,6,8,10)(7,11,12,9)(13,14,19,24)(15,20,26,35)(16,21,29,38)(17,22,32,41)(18,23,30,39)(25,34,43,44)(27,36,31,40)(28,37,33,42);(5,7,8,12)(6,9,10,11)(13,15)(14,20)(16,27)(17,30)(18,32)(19,26)(21,36)(22,39)(23,41)(24,35)(25,28)(29,31)(33,43)(34,37)(38,40)(42,44);(2,4)(13,16,19,29)(14,21,24,38)(15,25,26,43)(17,31,32,27)(18,33,30,28)(20,34,35,44)(22,40,41,36)(23,42,39,37);(13,17)(14,22)(15,18)(16,25)(19,32)(20,23)(21,34)(24,41)(26,30)(27,33)(28,31)(29,43)(35,39)(36,42)(37,40)(38,44);(15,17)(18,30)(20,22)(23,39)(25,31)(26,32)(27,43)(28,33)(34,40)(35,41)(36,44)(37,42)Polycyclic presentation:
2^9|a^2=fi,b^2=f,c^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, 6481690);