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 | #39131 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1598 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)(17,18,22,24)(19,28,29,40)(20,32,34,26)(21,27,23,37)(25,35,38,31)(30,45,44,43)(33,42,36,41)(39,47,48,46);(1,3)(2,5)(4,7)(6,8)(9,11)(10,13)(12,16)(14,15)(18,24)(19,29)(26,32)(27,37)(30,44)(31,35)(41,42)(46,47);(1,3)(2,4)(5,7)(6,8)(11,16)(13,15)(17,19,21,35,22,29,23,31)(18,25,37,40,24,38,27,28)(20,30,36,46,34,44,33,47)(26,39,41,45,32,48,42,43);(1,4,6,5)(2,3,7,8)(10,14)(13,15)(17,20)(18,26)(19,30)(21,36)(22,34)(23,33)(24,32)(25,39)(27,42)(28,43)(29,44)(31,47)(35,46)(37,41)(38,48)(40,45);(1,5,6,4)(2,3,7,8)(17,21,22,23)(18,27,24,37)(19,31,29,35)(20,33,34,36)(25,40,38,28)(26,41,32,42)(30,46,44,47)(39,43,48,45)Polycyclic presentation:
2^9|a^2=fhi,c^2=gi,d^2=fh,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^c=ei,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6897236);