Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | #36670 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1474 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)(4,6)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18,21,26,22,27,34,42)(19,23,30,36,29,25,40,41)(20,24,32,38,33,39,45,48)(28,35,43,46,31,37,44,47);(1,3)(2,5)(4,8)(6,7)(9,11,12,15)(10,13,14,16)(17,19)(18,23)(20,31)(21,30)(22,29)(24,37)(25,27)(26,36)(28,33)(32,44)(34,40)(35,39)(38,47)(41,42)(43,45)(46,48);(10,14)(11,15)(17,20,21,32,22,33,34,45)(18,24,26,38,27,39,42,48)(19,28,30,43,29,31,40,44)(23,35,36,46,25,37,41,47);(3,8)(5,7)(10,14)(13,16)(17,19)(18,25)(20,31)(21,30)(22,29)(23,27)(24,35)(26,41)(28,33)(32,44)(34,40)(36,42)(37,39)(38,46)(43,45)(47,48);(2,6)(5,7)(19,29)(23,25)(28,31)(30,40)(35,37)(36,41)(43,44)(46,47)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=h,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cgi,d^b=df,e^b=ei,d^c=di,e^d=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6319809);