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 | C22×2−1+4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2202 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)(9,10,12,14)(11,13,15,16)(17,18)(19,28)(20,25)(21,26)(22,36)(23,27)(24,31)(29,43)(30,38)(32,42)(33,39)(34,40)(35,41)(37,45)(44,48)(46,47);(1,3,4,8)(2,5,6,7)(9,11)(10,13)(12,15)(14,16)(20,33)(25,39)(29,44)(34,35)(37,47)(40,41)(43,48)(45,46);(11,15)(13,16)(17,19,23,32)(18,24,27,38)(20,29,33,44)(21,31,22,30)(25,37,39,47)(26,42,36,28)(34,46,35,45)(40,43,41,48);(2,6)(5,7)(17,20)(18,25)(19,29)(21,35)(22,34)(23,33)(24,37)(26,41)(27,39)(28,43)(30,46)(31,45)(32,44)(36,40)(38,47)(42,48);(3,8)(5,7)(11,15)(13,16)(17,21,23,22)(18,26,27,36)(19,30,32,31)(20,34,33,35)(24,28,38,42)(25,40,39,41)(29,45,44,46)(37,48,47,43)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,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, 6653061);