Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500669 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2023 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)(11,13)(12,14)(15,16)(17,18,22,25)(19,23,30,36)(20,24,31,39)(21,26,33,40)(27,34,29,37)(28,35,42,47)(32,38,44,48)(41,45,43,46);(1,3,5,2)(4,7,8,6)(9,11,12,15)(10,13,14,16)(17,19)(18,23)(20,28)(21,27)(22,30)(24,35)(25,36)(26,34)(29,33)(31,42)(32,41)(37,40)(38,45)(39,47)(43,44)(46,48);(1,4,5,8)(2,6,3,7)(10,14)(13,16)(19,27)(21,33)(23,34)(26,40)(28,41)(29,30)(32,44)(35,45)(36,37)(38,48)(42,43)(46,47);(2,3)(6,7)(10,14)(13,16)(17,20)(18,24)(19,28)(21,32)(22,31)(23,35)(25,39)(26,38)(27,41)(29,43)(30,42)(33,44)(34,45)(36,47)(37,46)(40,48);(18,25)(20,31)(23,36)(26,40)(28,42)(32,44)(34,37)(41,43)Polycyclic presentation:
2^9|a^2=i,b^2=fg,c^2=f,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, 7411835);