Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7309060 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2397 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1610 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,5)(3,6,7,8)(17,18,23,37,22,28,21,27)(19,24,30,40,32,42,31,41)(20,26,34,25,36,44,35,45)(29,39,48,38,33,46,47,43);(2,5)(3,7)(10,14)(11,12)(17,19,22,32)(18,24,28,42)(20,33,36,29)(21,31,23,30)(25,43,45,38)(26,46,44,39)(27,41,37,40)(34,47,35,48);(9,10)(11,16)(12,15)(13,14)(17,20,22,36)(18,25,28,45)(19,29,32,33)(21,34,23,35)(24,38,42,43)(26,27,44,37)(30,47,31,48)(39,41,46,40);(1,3,4,7)(2,6,5,8)(9,11,13,12)(10,15,14,16)(17,20)(18,26)(19,29)(21,35)(22,36)(23,34)(24,39)(25,37)(27,45)(28,44)(30,48)(31,47)(32,33)(38,40)(41,43)(42,46);(2,5)(6,8)(9,12,13,11)(10,15,14,16)(17,21,22,23)(18,27,28,37)(19,30,32,31)(20,34,36,35)(24,40,42,41)(25,44,45,26)(29,47,33,48)(38,39,43,46)Polycyclic presentation:
2^9|a^2=fgi,b^2=i,c^2=i,d^2=fhi,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dfh,e^b=eh,d^c=dh,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7397425);