Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470395 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1491 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)(3,8)(4,6)(5,7)(11,15)(14,16)(17,18,23,28)(19,29,34,45)(20,25,35,42)(21,26,36,43)(22,37,30,27)(24,38,41,33)(31,44,46,40)(32,39,47,48);(9,10)(11,14)(12,13)(15,16)(17,19,30,38,23,34,22,33)(18,24,37,45,28,41,27,29)(20,31,36,47,35,46,21,32)(25,39,26,40,42,48,43,44);(1,3,5,4)(2,6,7,8)(10,13)(14,16)(17,20)(18,25)(19,31)(21,22)(23,35)(24,39)(26,37)(27,43)(28,42)(29,44)(30,36)(32,33)(34,46)(38,47)(40,45)(41,48);(1,4,5,3)(2,6,7,8)(10,13)(14,16)(17,21)(18,26)(19,32)(20,30)(22,35)(23,36)(24,40)(25,27)(28,43)(29,39)(31,38)(33,46)(34,47)(37,42)(41,44)(45,48);(9,11,12,15)(10,14,13,16)(20,35)(21,36)(25,42)(26,43)(31,46)(32,47)(39,48)(40,44)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=g,d^2=g,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,d^c=di,e^c=ei,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6403197);