Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667932 |
| 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 | C2×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1572 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)(3,7)(4,6)(5,8)(17,18)(19,29)(20,33)(21,28)(22,24)(23,27)(25,32)(26,34)(30,42)(31,45)(35,40)(36,41)(37,38)(39,46)(43,44)(47,48);(1,3,4,8)(2,5,6,7)(9,10)(11,15)(12,13)(14,16)(18,24)(19,30)(20,34)(27,28)(32,37)(35,36)(39,48)(43,45);(3,8)(5,7)(10,13)(14,15)(17,19,22,30)(18,25,24,38)(20,31,34,44)(21,32,23,37)(26,39,33,48)(27,29,28,42)(35,46,36,47)(40,45,41,43);(2,6)(5,7)(17,20)(18,26)(19,31)(21,36)(22,34)(23,35)(24,33)(25,39)(27,41)(28,40)(29,43)(30,44)(32,47)(37,46)(38,48)(42,45);(9,11,12,16)(10,14,13,15)(17,21,22,23)(18,27,24,28)(19,32,30,37)(20,35,34,36)(25,29,38,42)(26,40,33,41)(31,46,44,47)(39,45,48,43)Polycyclic presentation:
2^9|b^2=fi,c^2=i,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=efh,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, 6699857);