Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500416 |
| 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 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1644 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)(3,6)(4,5)(7,8)(17,18)(19,24)(20,31)(21,26)(22,36)(23,27)(25,33)(28,43)(29,38)(30,39)(32,42)(34,40)(35,41)(37,45)(44,47)(46,48);(2,5)(6,8)(9,10)(11,15)(12,14)(13,16)(17,19)(18,24)(20,28)(21,34)(22,30)(23,29)(25,37)(26,40)(27,38)(31,43)(32,46)(33,45)(35,44)(36,39)(41,47)(42,48);(9,11)(10,13)(12,16)(14,15)(17,20,22,33,23,32,21,35)(18,25,26,31,27,41,36,42)(19,28,30,45,29,46,34,44)(24,37,40,43,38,47,39,48);(1,3,4,7)(2,6,5,8)(11,16)(13,15)(19,29)(20,32)(24,38)(25,41)(30,34)(31,42)(33,35)(39,40);(10,14)(13,15)(17,21,23,22)(18,26,27,36)(19,30,29,34)(20,33,32,35)(24,39,38,40)(25,42,41,31)(28,44,46,45)(37,43,47,48)Polycyclic presentation:
2^9|c^2=g,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=ehi,d^c=dhi,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6414542);