Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500669 |
| 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 | |
1568 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)(3,5)(4,6)(7,8)(9,10)(11,15)(12,14)(13,16)(17,18)(19,24)(20,30)(21,23)(22,26)(25,31)(27,39)(28,42)(29,35)(32,45)(33,36)(34,37)(38,44)(40,46)(41,47)(43,48);(1,3,5,2)(4,7,8,6)(9,11,12,16)(10,13,14,15)(18,23)(19,27)(20,31)(22,33)(24,37)(28,43)(29,34)(35,39)(38,48)(40,45)(41,44)(42,47);(10,14)(13,15)(17,19,21,29)(18,24,23,35)(20,28,31,44)(22,34,33,27)(25,38,30,42)(26,37,36,39)(32,43,46,41)(40,47,45,48);(2,3)(6,7)(11,16)(13,15)(17,20)(18,25)(19,28)(21,31)(22,32)(23,30)(24,38)(26,40)(27,41)(29,44)(33,46)(34,43)(35,42)(36,45)(37,47)(39,48);(1,4,5,8)(2,6,3,7)(18,23)(20,31)(24,35)(26,36)(28,44)(32,46)(37,39)(41,43)Polycyclic presentation:
2^9|b^2=fi,c^2=i,e^2=fgi,h^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,c^b=ch,d^b=df,h^b=hi,h^c=hi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7419841);