Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4465744 |
| 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, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1404 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,3)(4,7,8,6)(11,15)(13,16)(20,27)(24,29)(25,31)(30,32)(33,34)(35,39)(36,40)(37,38)(41,43)(42,47)(44,48)(45,46);(1,2)(3,5)(4,7)(6,8)(9,10,12,14)(11,13,15,16)(18,22)(23,28)(24,29)(30,32)(34,37)(35,41)(40,45)(42,44);(2,3)(6,7)(17,18,21,22)(19,23,26,28)(20,24,27,29)(25,30,31,32)(33,35,38,41)(34,39,37,43)(36,44,46,42)(40,48,45,47);(1,3,5,2)(4,7,8,6)(9,11)(10,13)(12,15)(14,16)(17,19,21,26)(18,23,22,28)(20,25,27,31)(24,30,29,32)(33,36,38,46)(34,40,37,45)(35,42,41,44)(39,47,43,48);(1,4)(2,6)(3,7)(5,8)(17,20)(18,24)(19,25)(21,27)(22,29)(23,30)(26,31)(28,32)(33,37,38,34)(35,43,41,39)(36,45,46,40)(42,48,44,47)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=gh,d^2=h,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^b=ef,d^c=df68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4385292);