Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514020 |
| 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 | #392 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1286 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,8)(3,7,4,6)(9,10,13,15)(11,14,16,12)(17,18,22,26)(19,23,30,38)(20,24,32,40)(21,25,34,42)(27,35,31,39)(28,36,29,37)(33,41,45,48)(43,46,44,47);(1,3,5,4)(2,6,8,7)(9,11)(10,14)(12,15)(13,16)(17,19,22,30)(18,23,26,38)(20,31,32,27)(21,28,34,29)(24,39,40,35)(25,36,42,37)(33,44,45,43)(41,47,48,46);(1,3,5,4)(2,7,8,6)(10,15)(11,16)(17,20)(18,24)(19,27)(21,33)(22,32)(23,35)(25,41)(26,40)(28,43)(29,44)(30,31)(34,45)(36,46)(37,47)(38,39)(42,48);(1,3,5,4)(2,7,8,6)(9,12,13,14)(10,16,15,11)(19,28)(21,34)(23,36)(25,42)(27,43)(29,30)(31,44)(33,45)(35,46)(37,38)(39,47)(41,48);(1,4,5,3)(2,7,8,6)(20,32)(24,40)(27,31)(33,45)(35,39)(41,48)(43,44)(46,47)Polycyclic presentation:
2^9|a^2=fgi,b^2=fi,c^2=f,d^2=fg,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cgi,d^b=dh,h^b=hi,e^c=ei,h^d=hi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7467291);