Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6368766 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C42 | C25 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1501 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,8)(4,9)(5,7)(6,11)(10,13)(12,15)(14,16)(17,18)(19,27)(20,31)(21,35)(22,25)(23,29)(24,34)(26,38)(28,41)(30,43)(32,39)(33,42)(36,40)(37,44)(45,46)(47,48);(1,3,4,11)(2,6,9,8)(5,12,13,16)(7,14,10,15)(17,19,22,30)(18,23,25,40)(20,32,34,47)(21,36,38,29)(24,41,31,45)(26,43,35,27)(28,37,46,33)(39,44,48,42);(2,7)(5,13)(6,14)(8,15)(9,10)(12,16)(17,18)(19,23)(20,24)(21,26)(22,25)(27,29)(28,39)(30,40)(31,34)(32,41)(33,42)(35,38)(36,43)(37,44)(45,47)(46,48);(1,3)(2,8)(4,11)(5,12)(6,9)(7,15)(10,14)(13,16)(17,20,22,34)(18,24,25,31)(19,28,30,46)(21,37,38,33)(23,39,40,48)(26,44,35,42)(27,45,43,41)(29,47,36,32);(3,11)(6,8)(12,16)(14,15)(18,25)(23,40)(24,31)(26,35)(27,43)(39,48)(41,45)(42,44)Polycyclic presentation:
2^9|b^2=hi,d^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=eh,f^a=fh,g^a=gi,d^b=df,e^b=ei,f^b=fh,e^c=eh,g^c=gi,e^d=ei,f^d=fh65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6275991);