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 | C2×#548 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1477 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)(4,6)(5,8)(9,10,12,14)(11,13,15,16)(17,18)(19,25)(20,33)(21,36)(22,28)(23,29)(24,30)(26,35)(27,39)(31,47)(32,42)(34,43)(37,44)(38,46)(40,45)(41,48);(1,3,5,4)(2,6,8,7)(9,11)(10,13)(12,15)(14,16)(17,19,23,32)(18,25,29,42)(20,24,35,22)(21,37,39,34)(26,28,33,30)(27,43,36,44)(31,40,48,38)(41,46,47,45);(1,3,5,4)(2,7,8,6)(10,14)(11,15)(17,20)(18,26)(19,22)(21,37)(23,35)(24,32)(25,30)(27,44)(28,42)(29,33)(31,40)(34,39)(36,43)(38,48)(41,45)(46,47);(1,4,5,3)(2,6,8,7)(10,14)(13,16)(17,21)(18,27)(19,31)(20,34)(22,38)(23,39)(24,40)(25,41)(26,43)(28,45)(29,36)(30,46)(32,48)(33,44)(35,37)(42,47);(2,8)(6,7)(17,22,23,24)(18,28,29,30)(19,20,32,35)(21,38,39,40)(25,33,42,26)(27,45,36,46)(31,34,48,37)(41,44,47,43)Polycyclic presentation:
2^9|a^2=gi,b^2=fi,c^2=f,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,d^c=di,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, 7433708);