Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500669 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#533 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1716 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,7)(3,6,4,8)(10,13)(11,15)(17,18,22,27)(19,28,30,24)(20,33,36,25)(21,37,23,26)(29,39,46,45)(31,40,32,44)(34,41,47,48)(35,42,38,43);(1,2)(3,6)(4,8)(5,7)(9,10)(11,14)(12,13)(15,16)(17,19,22,30)(18,24,27,28)(20,34,36,47)(21,32,23,31)(25,41,33,48)(26,44,37,40)(29,38,46,35)(39,42,45,43);(1,3,5,4)(2,6,7,8)(9,11,12,15)(10,14,13,16)(17,20,22,36)(18,25,27,33)(19,29,30,46)(21,35,23,38)(24,39,28,45)(26,42,37,43)(31,34,32,47)(40,48,44,41);(1,4,5,3)(2,6,7,8)(10,13)(14,16)(19,30)(24,28)(29,46)(31,32)(34,47)(39,45)(40,44)(41,48);(17,21,22,23)(18,26,27,37)(19,31,30,32)(20,35,36,38)(24,40,28,44)(25,42,33,43)(29,34,46,47)(39,48,45,41)Polycyclic presentation:
2^9|a^2=fgi,b^2=i,c^2=f,d^2=fg,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,c^b=ch,d^b=df,e^b=ei,h^b=hi,h^c=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7417693);