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 | #35210 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1627 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)(3,4)(5,7)(6,8)(9,10)(11,15)(12,14)(13,16)(17,18,23,27)(19,24,30,39)(20,25,33,43)(21,34,22,26)(28,37,46,47)(29,40,35,38)(31,41,45,48)(32,44,36,42);(1,3)(2,6)(4,7)(5,8)(9,11)(10,13)(12,16)(14,15)(17,19,23,30)(18,24,27,39)(20,31,33,45)(21,29,22,35)(25,41,43,48)(26,38,34,40)(28,36,46,32)(37,42,47,44);(10,14)(13,15)(17,20)(18,25)(19,28)(21,32)(22,36)(23,33)(24,37)(26,42)(27,43)(29,45)(30,46)(31,35)(34,44)(38,41)(39,47)(40,48);(2,7)(3,8)(11,16)(13,15)(17,21,23,22)(18,26,27,34)(19,29,30,35)(20,32,33,36)(24,38,39,40)(25,42,43,44)(28,45,46,31)(37,41,47,48);(1,4,5,6)(2,3,7,8)(18,27)(24,39)(25,43)(26,34)(37,47)(38,40)(41,48)(42,44)Polycyclic presentation:
2^9|a^2=i,b^2=i,d^2=i,e^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=dfgi,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, 7419134);