Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500186 |
| 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×#547 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1455 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)(3,6)(4,8)(5,7)(17,18,23,35,22,27,20,25)(19,28,29,24,32,37,31,39)(21,26,36,46,34,41,33,40)(30,42,43,38,44,47,45,48);(1,3,5,4)(2,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,24)(20,29)(21,30)(22,32)(23,31)(25,37)(26,38)(27,39)(28,35)(33,43)(34,44)(36,45)(40,47)(41,48)(42,46);(2,7)(6,8)(10,13)(11,15)(17,20,22,23)(18,25,27,35)(19,29,32,31)(21,33,34,36)(24,37,39,28)(26,40,41,46)(30,43,44,45)(38,47,48,42);(9,11,12,15)(10,14,13,16)(17,21,22,34)(18,26,27,41)(19,30,32,44)(20,33,23,36)(24,38,39,48)(25,40,35,46)(28,42,37,47)(29,43,31,45);(1,4,5,3)(2,8,7,6)(10,13)(14,16)(21,34)(26,41)(30,44)(33,36)(38,48)(40,46)(42,47)(43,45)Polycyclic presentation:
2^9|a^2=fi,b^2=g,c^2=i,d^2=hi,e^2=g,f^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,f^b=fi,d^c=dh,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6431738);