Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470894 |
| 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 | #2407 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1382 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)(4,8)(5,6)(9,10)(11,13)(12,14)(15,16)(17,18)(19,28)(20,31)(21,24)(22,26)(23,37)(25,33)(27,36)(29,43)(30,39)(32,46)(34,45)(35,42)(38,48)(40,44)(41,47);(2,6)(7,8)(9,11)(10,13)(12,15)(14,16)(17,19)(18,24)(20,29)(21,22)(23,30)(25,38)(26,28)(27,39)(31,45)(32,34)(33,44)(35,36)(37,42)(40,41)(43,46)(47,48);(1,3,5,4)(2,7,6,8)(11,15)(13,16)(17,20,23,33,22,32,36,47)(18,25,27,31,26,41,37,46)(19,29,30,44,21,34,35,48)(24,38,39,45,28,40,42,43);(11,15)(13,16)(17,21,22,19)(18,24,26,28)(20,29,32,34)(23,35,36,30)(25,40,41,38)(27,39,37,42)(31,43,46,45)(33,44,47,48);(1,4,5,3)(2,8,6,7)(10,14)(13,16)(19,21)(20,32)(24,28)(25,41)(30,35)(31,46)(33,47)(39,42)Polycyclic presentation:
2^9|c^2=fgi,d^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dhi,e^b=ei,d^c=di,e^c=ei,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6382466);