Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6668044 |
| 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×#449 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2285 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1760 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)(3,6)(4,5)(7,8)(9,10)(11,15)(12,14)(13,16)(17,18,23,26,22,27,21,28)(19,29,31,40,32,24,33,43)(20,34,36,25,37,44,38,42)(30,45,47,41,35,48,46,39);(2,5)(3,7)(9,11,12,16)(10,13,14,15)(17,19)(18,24)(20,35)(21,33)(22,32)(23,31)(25,41)(26,43)(27,29)(28,40)(30,37)(34,45)(36,46)(38,47)(39,42)(44,48);(11,16)(13,15)(17,20)(18,25)(19,30)(21,36)(22,37)(23,38)(24,39)(26,34)(27,42)(28,44)(29,41)(31,46)(32,35)(33,47)(40,45)(43,48);(2,5)(6,8)(10,14)(13,15)(17,21,22,23)(18,26,27,28)(19,31,32,33)(20,36,37,38)(24,40,29,43)(25,34,42,44)(30,46,35,47)(39,45,41,48);(1,3,4,7)(2,6,5,8)(18,27)(24,29)(25,42)(26,28)(34,44)(39,41)(40,43)(45,48)Polycyclic presentation:
2^9|a^2=gi,b^2=hi,d^2=i,e^2=fh,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, 6709743);