Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438975 |
| 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 | #2535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1557 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)(9,10)(11,16)(12,14)(13,15)(17,18)(19,24)(20,33)(21,26)(22,38)(23,28)(25,35)(27,30)(29,43)(31,40)(32,44)(34,39)(36,47)(37,48)(41,45)(42,46);(2,5)(6,8)(11,12)(14,16)(17,19)(18,24)(20,29)(21,30)(22,32)(23,31)(25,39)(26,27)(28,40)(33,43)(34,35)(36,45)(37,46)(38,44)(41,47)(42,48);(9,11,13,12)(10,14,15,16)(17,20,22,35,23,36,30,46)(18,25,27,33,28,42,38,47)(19,29,32,34,31,45,21,37)(24,39,26,43,40,48,44,41);(9,12,13,11)(10,14,15,16)(17,21,23,32)(18,26,28,44)(19,30,31,22)(20,34,36,37)(24,27,40,38)(25,41,42,43)(29,35,45,46)(33,39,47,48);(1,3,4,7)(2,6,5,8)(19,31)(21,32)(24,40)(26,44)(29,45)(34,37)(39,48)(41,43)Polycyclic presentation:
2^9|c^2=g,d^2=hi,e^2=f,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,e^b=ei,d^c=di,e^d=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6370731);