Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6343394 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×C22⋊C4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | #62805 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
2854 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,11,13,16)(4,9)(5,10)(6,8)(7,12,15,14)(17,18)(19,27,30,23)(20,24)(21,33)(22,26)(25,34)(28,39,41,35)(29,36,43,40)(31,37)(32,45)(38,44)(42,46,48,47);(1,3)(2,7)(4,12)(5,14)(6,13)(8,15)(9,11)(10,16)(17,19,22,30)(18,23,26,27)(20,28,31,41)(21,29,34,43)(24,35,37,39)(25,36,33,40)(32,42,44,48)(38,46,45,47);(2,8)(7,15)(9,10)(11,16)(17,20)(18,24)(19,28)(21,32)(22,31)(23,35)(25,38)(26,37)(27,39)(29,42)(30,41)(33,45)(34,44)(36,46)(40,47)(43,48);(20,31)(24,37)(28,41)(32,44)(35,39)(38,45)(42,48)(46,47);(1,4,6,5)(2,9,8,10)(3,12,13,14)(7,11,15,16)(17,21,22,34)(18,25,26,33)(19,29,30,43)(20,32,31,44)(23,36,27,40)(24,38,37,45)(28,42,41,48)(35,46,39,47)Polycyclic presentation:
2^9|a^2=g,b^2=h,e^2=hi,f^2=i,b^a=bf,c^a=ci,e^a=eh,f^a=fh,f^b=fi,g^b=ghi,d^c=dh92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6250601);