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 | #37656 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1790 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)(19,28)(20,32)(21,27)(22,24)(23,37)(25,30)(26,35)(29,46)(31,41)(33,45)(34,44)(36,43)(38,39)(40,47)(42,48);(2,6)(4,7)(9,11)(10,13)(12,16)(14,15)(18,24)(25,39)(26,32)(27,37)(28,41)(40,42)(43,44)(45,46);(1,3)(2,7)(4,6)(5,8)(11,16)(13,15)(17,19,22,31)(18,25,24,39)(20,33,35,29)(21,30,23,38)(26,42,32,40)(27,41,37,28)(34,48,36,47)(43,46,44,45);(2,6)(3,8)(10,14)(13,15)(17,20,22,35)(18,26,24,32)(19,29,31,33)(21,36,23,34)(25,40,39,42)(27,44,37,43)(28,45,41,46)(30,48,38,47);(1,4,5,7)(2,3,6,8)(17,21,22,23)(18,27,24,37)(19,30,31,38)(20,34,35,36)(25,41,39,28)(26,43,32,44)(29,47,33,48)(40,45,42,46)Polycyclic presentation:
2^9|c^2=i,d^2=i,e^2=fh,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, 6870859);