Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470922 |
| 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×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2280 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1645 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)(19,29)(20,34)(21,26)(22,27)(23,28)(24,31)(25,37)(30,38)(32,40)(33,39)(35,41)(36,42)(43,46)(44,47)(45,48);(2,5)(6,8)(9,11)(10,13)(12,15)(14,16)(17,19,22,32,23,33,21,31)(18,24,26,39,28,40,27,29)(20,30,37,46,36,45,35,44)(25,38,34,47,41,48,42,43);(1,3,4,7)(2,6,5,8)(11,15)(13,16)(17,20,21,35,23,36,22,37)(18,25,27,42,28,41,26,34)(19,30,31,44,33,45,32,46)(24,38,29,43,40,48,39,47);(11,15)(13,16)(17,21,23,22)(18,26,28,27)(19,31,33,32)(20,35,36,37)(24,39,40,29)(25,34,41,42)(30,44,45,46)(38,47,48,43);(10,14)(13,16)(20,36)(25,41)(30,45)(34,42)(35,37)(38,48)(43,47)(44,46)Polycyclic presentation:
2^9|b^2=g,c^2=f,d^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=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6380118);