Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533123 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2295 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1366 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,8)(4,6)(5,7)(10,13)(14,16)(17,18,23,27,22,26,21,35)(19,28,30,39,31,24,36,40)(20,32,33,41,34,42,37,25)(29,44,47,38,46,48,45,43);(1,3,5,4)(2,6,7,8)(9,10,12,13)(11,14,15,16)(17,19)(18,24)(20,29)(21,36)(22,31)(23,30)(25,38)(26,28)(27,40)(32,48)(33,47)(34,46)(35,39)(37,45)(41,43)(42,44);(10,13)(14,16)(17,20,22,34)(18,25,26,41)(19,29,31,46)(21,33,23,37)(24,38,28,43)(27,42,35,32)(30,45,36,47)(39,48,40,44);(1,4,5,3)(2,6,7,8)(11,15)(14,16)(18,26)(24,28)(25,41)(27,35)(32,42)(38,43)(39,40)(44,48);(9,11,12,15)(10,14,13,16)(17,21,22,23)(18,27,26,35)(19,30,31,36)(20,33,34,37)(24,39,28,40)(25,42,41,32)(29,45,46,47)(38,48,43,44)Polycyclic presentation:
2^9|a^2=g,b^2=fi,c^2=i,d^2=fhi,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,c^b=ch,e^b=ei,g^c=gi,e^d=eh71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6454668);