Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500669 |
| 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 | #35212 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1554 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)(4,7)(5,8)(10,13)(11,15)(17,18,22,27)(19,28,31,24)(20,25,35,42)(21,36,23,26)(29,45,46,38)(30,39,32,44)(33,48,47,40)(34,43,37,41);(1,3,6,2)(4,8,5,7)(9,10,12,13)(11,14,15,16)(17,19,22,31)(18,24,27,28)(20,33,35,47)(21,32,23,30)(25,40,42,48)(26,44,36,39)(29,37,46,34)(38,41,45,43);(1,4,6,5)(2,7,3,8)(9,11,12,15)(10,14,13,16)(17,20,22,35)(18,25,27,42)(19,29,31,46)(21,34,23,37)(24,38,28,45)(26,41,36,43)(30,33,32,47)(39,48,44,40);(2,3)(7,8)(10,13)(14,16)(17,21,22,23)(18,26,27,36)(19,30,31,32)(20,34,35,37)(24,39,28,44)(25,41,42,43)(29,33,46,47)(38,48,45,40);(1,5,6,4)(2,8,3,7)(18,27)(24,28)(25,42)(26,36)(38,45)(39,44)(40,48)(41,43)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=f,d^2=i,e^2=fgi,h^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,c^b=ch,d^b=df,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, 7420543);