Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573615 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×2−1+4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2223 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,8)(4,6)(5,7)(9,10,12,14)(11,13,15,16)(17,18)(19,24)(20,31)(21,26)(22,36)(23,27)(25,34)(28,45)(29,39)(30,41)(32,48)(33,44)(35,40)(37,43)(38,47)(42,46);(1,3,5,4)(2,6,7,8)(9,11)(10,13)(12,15)(14,16)(17,19,23,30)(18,24,27,41)(20,32,33,28)(21,35,22,29)(25,42,43,38)(26,40,36,39)(31,48,44,45)(34,46,37,47);(11,15)(13,16)(17,20)(18,25)(19,28)(21,34)(22,37)(23,33)(24,38)(26,44)(27,43)(29,46)(30,32)(31,36)(35,47)(39,45)(40,48)(41,42);(1,4,5,3)(2,6,7,8)(9,11)(10,13)(12,15)(14,16)(20,33)(25,43)(28,32)(31,44)(34,37)(38,42)(45,48)(46,47);(11,15)(13,16)(17,21,23,22)(18,26,27,36)(19,29,30,35)(20,34,33,37)(24,39,41,40)(25,44,43,31)(28,46,32,47)(38,45,42,48)Polycyclic presentation:
2^9|a^2=hi,b^2=fi,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=eh,d^c=dh,g^c=gi,e^d=ehi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6500173);