Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6432838 |
| 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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1586 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)(10,14)(11,12)(17,18,23,26)(19,24,30,41)(20,33,36,44)(21,28,22,27)(25,37,43,35)(29,45,34,48)(31,40,32,39)(38,47,42,46);(2,5)(6,8)(9,10,13,14)(11,16,12,15)(17,19)(18,24)(20,34)(21,31)(22,32)(23,30)(25,42)(26,41)(27,39)(28,40)(29,36)(33,48)(35,47)(37,46)(38,43)(44,45);(9,11,13,12)(10,15,14,16)(17,20,23,36)(18,25,26,43)(19,29,30,34)(21,35,22,37)(24,38,41,42)(27,33,28,44)(31,46,32,47)(39,45,40,48);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,15,14,16)(18,26)(19,30)(25,43)(27,28)(29,34)(31,32)(33,44)(46,47);(17,21,23,22)(18,27,26,28)(19,31,30,32)(20,35,36,37)(24,39,41,40)(25,33,43,44)(29,46,34,47)(38,45,42,48)Polycyclic presentation:
2^9|a^2=i,b^2=hi,c^2=h,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ei,g^a=gi,c^b=ch,d^b=di,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6341954);