Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437770 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1606 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)(11,15)(12,16)(13,14)(17,18,24,40,23,29,22,28)(19,30,39,25,34,48,33,43)(20,26,21,27,36,45,37,46)(31,42,38,41,35,47,32,44);(10,14)(11,12)(17,19)(18,25)(20,35)(21,38)(22,39)(23,34)(24,33)(26,44)(27,47)(28,48)(29,43)(30,40)(31,36)(32,37)(41,45)(42,46);(2,5)(6,8)(9,11,13,12)(10,15,14,16)(17,20)(18,26)(19,31)(21,24)(22,37)(23,36)(25,41)(27,40)(28,46)(29,45)(30,42)(32,33)(34,35)(38,39)(43,44)(47,48);(9,12,13,11)(10,16,14,15)(17,21)(18,27)(19,32)(20,22)(23,37)(24,36)(25,42)(26,28)(29,46)(30,44)(31,39)(33,35)(34,38)(40,45)(41,48)(43,47);(1,3,4,7)(2,6,5,8)(20,36)(21,37)(26,45)(27,46)(31,35)(32,38)(41,44)(42,47)Polycyclic presentation:
2^9|a^2=fh,c^2=hi,d^2=hi,e^2=g,f^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,d^c=di,e^c=ei,e^d=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6375441);