Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18446 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C2×#386 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1050 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 48 points:
(1,2,6,9,7,3,5,10)(4,12,14,8,15,16,13,11)(17,18)(19,28,32,39)(20,33)(21,26)(22,37)(23,27)(24,31,40,30)(25,35)(29,44,47,38)(34,42)(36,41)(43,45,48,46);(1,3,7,2)(4,11,15,8)(5,10,6,9)(12,14,16,13)(17,19,22,31,23,32,21,30)(18,24,37,28,27,40,26,39)(20,29,34,45,36,47,35,46)(25,38,33,43,42,44,41,48);(1,4,7,15)(2,8,3,11)(5,14,6,13)(9,12,10,16)(17,20,23,36)(18,25,27,42)(19,29,32,47)(21,35,22,34)(24,38,40,44)(26,41,37,33)(28,43,39,48)(30,46,31,45);(1,5,7,6)(2,9,3,10)(4,13,15,14)(8,16,11,12)(17,21,23,22)(18,26,27,37)(19,30,32,31)(20,34,36,35)(24,39,40,28)(25,33,42,41)(29,45,47,46)(38,43,44,48)Polycyclic presentation:
2^9|a^2=g,b^2=egh,c^2=h,d^2=h,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,f^a=fhi,e^b=eh,f^b=fi,g^b=gh,d^c=dh,e^c=ei,f^c=fh,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 102604);