Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500120 |
| 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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1380 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)(3,8)(4,6)(5,7)(17,18,22,28,23,27,21,26)(19,24,30,38,31,39,35,41)(20,32,33,40,34,42,36,25)(29,43,45,48,46,47,44,37);(1,3,5,4)(2,6,7,8)(9,10)(11,15)(12,14)(13,16)(17,19)(18,24)(20,29)(21,35)(22,30)(23,31)(25,37)(26,41)(27,39)(28,38)(32,43)(33,45)(34,46)(36,44)(40,48)(42,47);(9,11)(10,13)(12,16)(14,15)(17,20,23,34)(18,25,27,40)(19,29,31,46)(21,33,22,36)(24,37,39,48)(26,32,28,42)(30,44,35,45)(38,47,41,43);(11,16)(13,15)(17,21,23,22)(18,26,27,28)(19,30,31,35)(20,33,34,36)(24,38,39,41)(25,32,40,42)(29,44,46,45)(37,47,48,43);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(18,27)(24,39)(25,40)(26,28)(32,42)(37,48)(38,41)(43,47)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=i,d^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=di,e^b=eh,d^c=dh,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6413301);