Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667932 |
| 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 | #2407 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1536 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,6)(3,7,8,5)(17,18,21,26)(19,28,30,23)(20,32,33,25)(22,36,35,27)(24,37,41,34)(29,39,45,43)(31,40,47,44)(38,46,42,48);(1,3)(2,5)(4,8)(6,7)(9,10)(11,14)(12,13)(15,16)(17,19,21,30)(18,23,26,28)(20,29,33,45)(22,31,35,47)(24,38,41,42)(25,39,32,43)(27,40,36,44)(34,46,37,48);(10,13)(14,16)(17,20)(18,24)(19,29)(21,33)(22,37)(23,38)(25,27)(26,41)(28,42)(30,45)(31,48)(32,36)(34,35)(39,40)(43,44)(46,47);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(17,20)(18,25)(19,29)(21,33)(22,34)(23,39)(24,27)(26,32)(28,43)(30,45)(31,46)(35,37)(36,41)(38,40)(42,44)(47,48);(3,8)(5,7)(10,13)(14,16)(20,33)(24,41)(25,32)(29,45)(34,37)(38,42)(39,43)(46,48)Polycyclic presentation:
2^9|a^2=f,b^2=i,d^2=h,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efhi,e^c=ei,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6674408);