Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682440 |
| 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)(17,18,22,27)(19,23,30,26)(20,31,33,37)(21,25,36,44)(24,39,41,28)(29,38,34,42)(32,47,35,48)(40,46,43,45);(1,3)(2,5)(4,8)(6,7)(9,10)(11,14)(12,13)(15,16)(17,19,22,30)(18,23,27,26)(20,28,33,39)(21,34,36,29)(24,37,41,31)(25,42,44,38)(32,46,35,45)(40,47,43,48);(2,6)(3,8)(10,13)(14,16)(17,20,30,39,22,33,19,28)(18,24,23,37,27,41,26,31)(21,35,34,45,36,32,29,46)(25,43,38,47,44,40,42,48);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(17,21,22,36)(18,25,27,44)(19,29,30,34)(20,32,33,35)(23,38,26,42)(24,40,41,43)(28,45,39,46)(31,47,37,48);(10,13)(14,16)(21,36)(25,44)(29,34)(32,35)(38,42)(40,43)(45,46)(47,48)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fgi,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,d^c=di,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6693630);