Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6668044 |
| 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×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1767 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,14)(12,13)(15,16)(17,18,22,28,23,27,21,26)(19,24,30,38,31,39,35,41)(20,32,33,40,34,42,36,25)(29,43,45,48,46,47,44,37);(1,3)(2,5)(4,8)(6,7)(10,13)(11,15)(17,19,23,31)(18,24,27,39)(20,29,34,46)(21,35,22,30)(25,37,40,48)(26,41,28,38)(32,43,42,47)(33,45,36,44);(9,11,12,15)(10,14,13,16)(17,20)(18,25)(19,29)(21,33)(22,36)(23,34)(24,37)(26,32)(27,40)(28,42)(30,44)(31,46)(35,45)(38,47)(39,48)(41,43);(2,6)(5,7)(10,13)(14,16)(17,21,23,22)(18,26,27,28)(19,30,31,35)(20,33,34,36)(24,38,39,41)(25,32,40,42)(29,44,46,45)(37,47,48,43);(3,8)(5,7)(18,27)(24,39)(25,40)(26,28)(32,42)(37,48)(38,41)(43,47)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=h,d^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,d^b=di,e^b=efh,g^c=gi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6664490);