Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4861280 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1296 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,5)(6,8)(9,10)(11,12)(13,14)(15,16)(17,18,22,25)(19,26,23,20)(21,29,30,24)(27,28,31,32)(35,41)(36,43)(38,44)(39,46);(10,14)(11,16)(19,23)(20,26)(27,31)(28,32)(33,34,37,40)(35,38,41,44)(36,39,43,46)(42,45,47,48);(9,11)(10,15)(12,14)(13,16)(17,19,22,23)(18,20,25,26)(21,27,30,31)(24,28,29,32)(33,34)(35,38)(36,39)(37,40)(41,44)(42,45)(43,46)(47,48);(1,2)(3,6)(4,5)(7,8)(9,11)(10,12)(13,16)(14,15)(17,20)(18,23)(19,25)(21,28)(22,26)(24,31)(27,29)(30,32)(33,35,37,41)(34,38,40,44)(36,42,43,47)(39,45,46,48);(1,3,4,7)(2,6,5,8)(9,12)(10,16)(11,14)(13,15)(17,21,22,30)(18,24,25,29)(19,27,23,31)(20,28,26,32)(33,36)(34,39)(35,42)(37,43)(38,45)(40,46)(41,47)(44,48)Polycyclic presentation:
2^9|a^2=fg,b^2=gh,c^2=fg,d^2=gh,e^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4819117);