Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2760365 |
| 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 | |
1437 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(3,7)(6,8)(17,18,22,19)(20,23,27,25)(21,26,30,24)(28,32,29,31)(33,34)(35,38)(36,39)(37,40)(41,44)(42,46)(43,47)(45,48);(1,2,5,4)(3,6,7,8)(10,13)(14,16)(18,19)(20,27)(24,26)(28,29)(34,35)(39,41)(40,42)(47,48);(9,10)(11,14)(12,13)(15,16)(17,19,22,18)(20,23,27,25)(21,26,30,24)(28,31,29,32)(33,35,38,34)(36,39,44,41)(37,42,46,40)(43,47,45,48);(1,3)(2,6)(4,8)(5,7)(17,20)(18,23)(19,25)(21,29)(22,27)(24,32)(26,31)(28,30)(33,36,38,44)(34,39,35,41)(37,45,46,43)(40,48,42,47);(1,4,5,2)(3,8,7,6)(9,11,12,15)(10,14,13,16)(17,21)(18,24)(19,26)(20,28)(22,30)(23,31)(25,32)(27,29)(33,37,38,46)(34,40,35,42)(36,43,44,45)(39,47,41,48)Polycyclic presentation:
2^9|a^2=fg,b^2=i,c^2=f,d^2=g,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2705374);