Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6885065 |
| 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 | #38032 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1591 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)(3,6)(4,5)(7,8)(17,18,23,28,22,26,21,27)(19,29,32,41,31,24,36,39)(20,33,35,40,34,42,37,25)(30,44,47,38,46,48,45,43);(2,5)(3,7)(9,10)(11,14)(12,13)(15,16)(17,19)(18,24)(20,30)(21,36)(22,31)(23,32)(25,38)(26,29)(27,41)(28,39)(33,48)(34,46)(35,47)(37,45)(40,43)(42,44);(1,3,4,7)(2,6,5,8)(10,13)(11,15)(17,20,22,34)(18,25,26,40)(19,30,31,46)(21,35,23,37)(24,38,29,43)(27,33,28,42)(32,45,36,47)(39,44,41,48);(2,5)(6,8)(9,11,12,15)(10,14,13,16)(18,26)(19,31)(20,34)(27,28)(32,36)(35,37)(38,43)(44,48);(10,13)(14,16)(17,21,22,23)(18,27,26,28)(19,32,31,36)(20,35,34,37)(24,39,29,41)(25,33,40,42)(30,45,46,47)(38,44,43,48)Polycyclic presentation:
2^9|a^2=g,c^2=f,d^2=fhi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,d^b=di,e^b=efh,d^c=dfh,g^c=gi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6844072);