Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6884969 |
| 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×#537 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1502 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)(17,18,23,36,21,24,22,27)(19,28,35,25,29,41,31,38)(20,32,37,40,33,26,34,47)(30,44,48,46,43,42,45,39);(1,3,4,8)(2,5,6,7)(9,10)(11,15)(12,14)(13,16)(18,24)(19,29)(20,33)(27,36)(31,35)(34,37)(39,46)(42,44);(3,8)(5,7)(9,11)(10,13)(12,16)(14,15)(17,19,21,29)(18,25,24,38)(20,30,33,43)(22,35,23,31)(26,39,32,46)(27,41,36,28)(34,48,37,45)(40,44,47,42);(2,6)(5,7)(11,16)(13,15)(17,20)(18,26)(19,30)(21,33)(22,34)(23,37)(24,32)(25,39)(27,40)(28,42)(29,43)(31,45)(35,48)(36,47)(38,46)(41,44);(10,14)(13,15)(18,24)(20,33)(25,38)(27,36)(28,41)(30,43)(34,37)(45,48)Polycyclic presentation:
2^9|a^2=fg,b^2=fi,c^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,d^b=di,e^b=efh,d^c=dfh,g^c=gi,e^d=ei59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6801457);