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 | #3444 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1000 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 48 points:
(1,2,5,7,6,10,4,9)(3,11,12,8,14,15,13,16)(17,18)(19,29,32,39)(20,33)(21,27)(22,26)(23,28)(24,31,40,36)(25,34)(30,44,47,38)(35,41)(37,42)(43,45,48,46);(2,7)(4,5)(8,11)(9,10)(12,13)(15,16)(17,19,21,36,23,32,22,31)(18,24,27,39,28,40,26,29)(20,30,34,46,35,47,37,45)(25,38,41,48,42,44,33,43);(1,3)(2,8)(4,12)(5,13)(6,14)(7,11)(9,15)(10,16)(17,20)(18,25)(19,30)(21,34)(22,37)(23,35)(24,38)(26,33)(27,41)(28,42)(29,43)(31,45)(32,47)(36,46)(39,48)(40,44);(1,4,6,5)(2,9,10,7)(3,12,14,13)(8,15,16,11)(17,21,23,22)(18,26,28,27)(19,31,32,36)(20,34,35,37)(24,39,40,29)(25,33,42,41)(30,45,47,46)(38,48,44,43)Polycyclic presentation:
2^9|a^2=g,b^2=eg,d^2=h,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dhi,f^a=fhi,d^b=dh,e^b=eh,f^b=fi,g^b=gh,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, 102598);