Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6907945 |
| 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 | #38903 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1596 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,6,7)(3,4,8,5)(9,10)(11,15)(12,14)(13,16)(17,18,21,25)(19,27,29,23)(20,31,33,46)(22,36,35,26)(24,39,41,34)(28,43,32,47)(30,38,45,42)(37,44,40,48);(1,3)(2,5)(4,7)(6,8)(9,11)(10,13)(12,16)(14,15)(17,19,21,29)(18,23,25,27)(20,32,33,28)(22,30,35,45)(24,40,41,37)(26,38,36,42)(31,43,46,47)(34,48,39,44);(1,3)(2,4)(5,7)(6,8)(11,16)(13,15)(17,20,22,34,21,33,35,39)(18,24,26,31,25,41,36,46)(19,28,30,44,29,32,45,48)(23,37,38,47,27,40,42,43);(1,4,6,5)(2,3,7,8)(10,14)(13,15)(18,25)(19,29)(24,41)(26,36)(28,32)(30,45)(31,46)(44,48);(1,5,6,4)(2,3,7,8)(20,33)(24,41)(28,32)(31,46)(34,39)(37,40)(43,47)(44,48)Polycyclic presentation:
2^9|a^2=fhi,b^2=i,c^2=g,d^2=fh,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=dfh,e^c=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6897112);