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 | #2540 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1299 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)(11,16)(12,14)(13,15)(17,18,22,26)(19,28,32,24)(20,33,36,25)(21,37,23,27)(29,39,34,42)(30,40,31,41)(35,43,38,44)(45,48,46,47);(2,6)(7,8)(9,10)(11,16)(12,14)(13,15)(17,19,22,32)(18,24,26,28)(20,34,36,29)(21,31,23,30)(25,42,33,39)(27,41,37,40)(35,46,38,45)(43,48,44,47);(1,3,5,4)(2,7,6,8)(9,11,13,12)(10,14,15,16)(17,20)(18,25)(19,29)(21,38)(22,36)(23,35)(24,39)(26,33)(27,44)(28,42)(30,45)(31,46)(32,34)(37,43)(40,47)(41,48);(1,3,5,4)(2,7,6,8)(10,15)(14,16)(18,26)(19,30)(21,23)(24,40)(25,33)(28,41)(29,45)(31,32)(34,46)(35,38)(39,47)(42,48);(1,4,5,3)(2,7,6,8)(9,12,13,11)(10,14,15,16)(17,21,22,23)(18,27,26,37)(19,31,32,30)(20,35,36,38)(24,41,28,40)(25,43,33,44)(29,45,34,46)(39,47,42,48)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fg,d^2=fi,e^2=fgi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,e^c=ei,e^d=ei,h^d=hi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7489899);