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×#555 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1528 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)(11,15)(12,14)(13,16)(17,18,22,28)(19,25,31,24)(20,26,34,44)(21,35,23,27)(29,41,47,38)(30,39,36,40)(32,48,46,42)(33,45,37,43);(2,6)(7,8)(9,11)(10,13)(12,16)(14,15)(17,19,22,31)(18,24,28,25)(20,32,34,46)(21,30,23,36)(26,42,44,48)(27,39,35,40)(29,37,47,33)(38,43,41,45);(1,3,5,4)(2,7,6,8)(10,14)(11,16)(17,19)(18,25)(20,29)(21,36)(22,31)(23,30)(24,28)(26,41)(27,39)(32,33)(34,47)(35,40)(37,46)(38,44)(42,43)(45,48);(10,14)(13,15)(17,20,22,34)(18,26,28,44)(19,29,31,47)(21,33,23,37)(24,38,25,41)(27,43,35,45)(30,46,36,32)(39,42,40,48);(1,4,5,3)(2,7,6,8)(10,14)(13,15)(17,21,22,23)(18,27,28,35)(19,30,31,36)(20,33,34,37)(24,39,25,40)(26,43,44,45)(29,46,47,32)(38,42,41,48)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fgi,d^2=i,e^2=fg,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, 7489230);