Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806839 |
| 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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1253 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)(3,7)(4,8)(5,6)(9,10,12,14)(11,15,16,13)(17,18)(19,24)(20,32)(21,26)(22,27)(23,28)(25,31)(29,38)(30,39)(33,44)(34,41)(35,40)(36,42)(37,43)(45,47)(46,48);(9,11)(10,13)(12,16)(14,15)(17,19,22,30)(18,24,27,39)(20,33,31,23)(21,35,36,34)(25,28,32,44)(26,40,42,41)(29,45,46,37)(38,47,48,43);(2,6)(7,8)(11,16)(13,15)(17,20,23,19,22,31,33,30)(18,25,28,39,27,32,44,24)(21,35,37,46,36,34,45,29)(26,41,43,38,42,40,47,48);(1,3,5,4)(2,7,6,8)(10,14)(13,15)(21,36)(26,42)(29,46)(34,35)(37,45)(38,48)(40,41)(43,47);(1,4,5,3)(2,8,6,7)(17,21,23,37,22,36,33,45)(18,26,28,43,27,42,44,47)(19,29,31,35,30,46,20,34)(24,38,25,40,39,48,32,41)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=fh,d^2=gi,e^2=fghi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6751388);