Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438991 |
| 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 | |
1532 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,8)(3,7,4,6)(9,10,12,14)(11,13,15,16)(17,18)(19,28)(20,24)(21,25)(22,27)(23,33)(26,30)(29,41)(31,43)(32,37)(34,39)(35,40)(36,42)(38,45)(44,48)(46,47);(1,3,5,4)(2,6,8,7)(9,11)(10,13)(12,15)(14,16)(19,29)(20,34)(23,37)(24,39)(28,41)(32,33)(35,36)(40,42);(1,3,5,4)(2,7,8,6)(11,15)(13,16)(17,19,30,32,22,29,21,33)(18,23,25,41,27,37,26,28)(20,31,35,46,34,44,36,45)(24,38,42,48,39,47,40,43);(1,4,5,3)(2,6,8,7)(11,15)(13,16)(17,20)(18,24)(19,31)(21,36)(22,34)(23,38)(25,42)(26,40)(27,39)(28,43)(29,44)(30,35)(32,46)(33,45)(37,47)(41,48);(17,21,22,30)(18,25,27,26)(19,32,29,33)(20,35,34,36)(23,28,37,41)(24,40,39,42)(31,45,44,46)(38,48,47,43)Polycyclic presentation:
2^9|a^2=fhi,b^2=f,c^2=fgi,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,e^c=ei,e^d=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6365109);