Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470894 |
| 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×#535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1672 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)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,30)(20,35)(21,26)(22,27)(23,29)(24,33)(25,36)(28,31)(32,44)(34,42)(37,46)(38,45)(39,47)(40,41)(43,48);(2,5)(6,8)(9,11)(10,13)(12,15)(14,16)(17,19,31,48,23,34,22,33)(18,24,27,42,29,43,28,30)(20,21,32,38,37,40,36,39)(25,41,46,45,44,26,35,47);(1,3,4,7)(2,6,5,8)(11,15)(13,16)(17,20,22,36,23,37,31,32)(18,25,28,35,29,44,27,46)(19,21,33,39,34,40,48,38)(24,41,30,47,43,26,42,45);(11,15)(13,16)(17,21)(18,26)(19,32)(20,33)(22,39)(23,40)(24,35)(25,42)(27,47)(28,45)(29,41)(30,44)(31,38)(34,36)(37,48)(43,46);(10,14)(13,16)(17,22,23,31)(18,27,29,28)(19,33,34,48)(20,32,37,36)(21,38,40,39)(24,42,43,30)(25,35,44,46)(26,45,41,47)Polycyclic presentation:
2^9|b^2=gi,c^2=f,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,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, 6380093);