Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6343406 |
| 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 | |
2857 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)(3,10,12,15)(4,13)(5,8)(6,9)(7,11,16,14)(17,18)(19,27,30,23)(20,24)(21,25)(22,26)(28,39,42,34)(29,40,43,35)(31,36)(32,37)(33,38)(41,47,48,45)(44,46);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,16)(10,13)(17,19,22,30)(18,23,26,27)(20,28,32,42)(21,29,33,43)(24,34,37,39)(25,35,38,40)(31,41,44,48)(36,45,46,47);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(17,20,22,32)(18,24,26,37)(19,28,30,42)(21,31,33,44)(23,34,27,39)(25,36,38,46)(29,41,43,48)(35,45,40,47);(17,21)(18,25)(19,29)(20,31)(22,33)(23,35)(24,36)(26,38)(27,40)(28,41)(30,43)(32,44)(34,45)(37,46)(39,47)(42,48);(21,33)(25,38)(29,43)(31,44)(35,40)(36,46)(41,48)(45,47)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=hi,f^2=i,b^a=bf,c^a=ci,f^a=fh,f^b=fi,g^b=ghi,e^d=eh92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6250610);