Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682360 |
| 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×#121 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1517 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)(3,7)(4,6)(5,8)(17,18,23,27)(19,24,22,36)(20,30,33,44)(21,26,35,41)(25,28,39,32)(29,37,34,40)(31,43,46,47)(38,42,48,45);(1,3)(2,5)(4,8)(6,7)(9,10)(11,14)(12,13)(15,16)(17,19,23,22)(18,24,27,36)(20,28,33,32)(21,34,35,29)(25,30,39,44)(26,40,41,37)(31,45,46,42)(38,47,48,43);(2,6)(3,8)(10,13)(14,16)(17,20,22,32,23,33,19,28)(18,25,24,30,27,39,36,44)(21,31,34,45,35,46,29,42)(26,38,37,43,41,48,40,47);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(17,21,23,35)(18,26,27,41)(19,29,22,34)(20,31,33,46)(24,37,36,40)(25,38,39,48)(28,42,32,45)(30,43,44,47);(10,13)(14,16)(21,35)(26,41)(29,34)(31,46)(37,40)(38,48)(42,45)(43,47)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fg,d^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6693498);