Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6499814 |
| 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 | C2×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1848 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,6)(3,5,7,8)(17,18)(19,28)(20,32)(21,27)(22,36)(23,26)(24,30)(25,33)(29,44)(31,40)(34,42)(35,41)(37,39)(38,46)(43,45)(47,48);(1,3,4,7)(2,5,6,8)(9,10)(11,14)(12,13)(15,16)(17,19,23,31)(18,24,26,39)(20,29,33,45)(21,30,22,37)(25,38,32,48)(27,40,36,28)(34,46,35,47)(41,43,42,44);(2,6)(5,8)(10,13)(11,15)(17,20)(18,25)(19,29)(21,35)(22,34)(23,33)(24,38)(26,32)(27,42)(28,43)(30,47)(31,45)(36,41)(37,46)(39,48)(40,44);(9,11,12,15)(10,14,13,16)(18,26)(20,33)(24,39)(27,36)(28,40)(29,45)(34,35)(46,47);(10,13)(14,16)(17,21,23,22)(18,27,26,36)(19,30,31,37)(20,34,33,35)(24,40,39,28)(25,41,32,42)(29,46,45,47)(38,43,48,44)Polycyclic presentation:
2^9|a^2=gi,b^2=g,d^2=h,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fi,c^b=ch,e^b=eh,f^b=fi,d^c=dhi,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6426633);