Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6907993 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#466 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2298 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1374 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)(3,6)(4,8)(5,7)(9,10,12,14)(11,15,16,13)(17,18)(19,28)(20,33)(21,27)(22,24)(23,37)(25,32)(26,35)(29,43)(30,45)(31,38)(34,41)(36,40)(39,47)(42,44)(46,48);(1,2,6,3)(4,7,5,8)(9,11,12,16)(10,13,14,15)(18,24)(19,29)(26,33)(27,37)(30,44)(31,32)(40,41)(46,47);(1,3,6,2)(4,7,5,8)(11,16)(13,15)(17,19,23,32,22,29,21,31)(18,25,27,28,24,38,37,43)(20,30,34,46,35,44,36,47)(26,39,41,42,33,48,40,45);(1,4,6,5)(2,7,3,8)(10,14)(13,15)(17,20,22,35)(18,26,24,33)(19,30,29,44)(21,36,23,34)(25,39,38,48)(27,41,37,40)(28,42,43,45)(31,47,32,46);(1,5,6,4)(2,7,3,8)(17,21,22,23)(18,27,24,37)(19,31,29,32)(20,34,35,36)(25,28,38,43)(26,40,33,41)(30,46,44,47)(39,45,48,42)Polycyclic presentation:
2^9|a^2=hi,b^2=fi,c^2=g,d^2=fhi,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=dfh,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6897402);