Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993947 |
| 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 | C2×#449 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1520 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)(11,15)(14,16)(17,18)(19,27)(20,33)(21,37)(22,25)(23,32)(24,35)(26,28)(29,45)(30,40)(31,39)(34,43)(36,42)(38,47)(41,46)(44,48);(2,5)(3,7)(10,13)(11,15)(17,19,28,30,22,31,21,32)(18,23,37,39,25,40,26,27)(20,34,44,46,35,29,36,47)(24,41,48,43,33,38,42,45);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(17,20)(18,24)(19,29)(21,36)(22,35)(23,38)(25,33)(26,42)(27,43)(28,44)(30,47)(31,34)(32,46)(37,48)(39,45)(40,41);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(19,30)(20,35)(21,28)(23,39)(24,33)(26,37)(27,40)(29,46)(31,32)(34,47)(38,43)(41,45);(9,11,12,15)(10,14,13,16)(18,25)(19,31)(20,35)(26,37)(30,32)(36,44)(38,41)(43,45)Polycyclic presentation:
2^9|b^2=fg,d^2=gi,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=df,e^b=eh,d^c=dh,e^c=ei,f^d=fi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7069254);