Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470421 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#121 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1684 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,6)(4,7)(5,8)(9,13)(10,14)(11,15)(12,16)(17,18,23,21)(19,27,30,45)(20,31,36,43)(22,38,39,26)(24,40,37,29)(25,34,33,35)(28,41,32,42)(44,46,48,47);(1,3)(2,6)(4,11)(5,9)(7,15)(8,13)(10,12)(14,16)(17,19,22,29,23,30,39,40)(18,24,26,27,21,37,38,45)(20,32,34,47,36,28,35,46)(25,42,43,48,33,41,31,44);(3,9)(6,13)(10,11)(14,15)(17,20)(18,25)(19,28)(21,33)(22,34)(23,36)(24,41)(26,43)(27,44)(29,46)(30,32)(31,38)(35,39)(37,42)(40,47)(45,48);(1,4,5,12)(2,7,8,16)(3,10,9,11)(6,14,13,15)(17,21,23,18)(19,24,30,37)(20,33,36,25)(22,38,39,26)(27,40,45,29)(28,41,32,42)(31,35,43,34)(44,47,48,46);(2,8)(6,13)(7,16)(14,15)(18,21)(24,37)(25,33)(26,38)(27,45)(31,43)(41,42)(44,48)Polycyclic presentation:
2^9|a^2=i,b^2=f,d^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,g^c=gi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6389100);