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 | #37659 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1785 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,4)(5,6)(7,8)(9,10)(11,15)(12,14)(13,16)(17,18,22,27)(19,28,23,24)(20,31,33,44)(21,34,36,26)(25,38,41,29)(30,39,37,43)(32,48,35,47)(40,45,42,46);(2,6)(4,7)(9,11)(10,13)(12,16)(14,15)(17,19,22,23)(18,24,27,28)(20,29,33,38)(21,30,36,37)(25,31,41,44)(26,39,34,43)(32,45,35,46)(40,47,42,48);(1,3)(2,7)(4,6)(5,8)(11,16)(13,15)(17,20,22,33)(18,25,27,41)(19,29,23,38)(21,35,36,32)(24,31,28,44)(26,42,34,40)(30,46,37,45)(39,48,43,47);(2,6)(3,8)(10,14)(13,15)(17,21)(18,26)(19,30)(20,32)(22,36)(23,37)(24,39)(25,40)(27,34)(28,43)(29,45)(31,47)(33,35)(38,46)(41,42)(44,48);(1,4,5,7)(2,3,6,8)(21,36)(26,34)(30,37)(32,35)(39,43)(40,42)(45,46)(47,48)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=i,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efhi,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, 6870882);