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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1378 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(3,7)(6,8)(17,18,22,19)(20,25,27,23)(21,26,30,24)(28,31,29,32)(33,34,37,40)(35,41,43,38)(36,39,44,46)(42,47,48,45);(1,2,5,4)(3,6,7,8)(10,14)(15,16)(18,19)(20,27)(24,26)(28,29)(36,44)(39,46)(42,48)(45,47);(9,10)(11,15)(12,16)(13,14)(17,19,22,18)(20,23,27,25)(21,26,30,24)(28,31,29,32)(33,35,37,43)(34,38,40,41)(36,42,44,48)(39,45,46,47);(1,3)(2,6)(4,8)(5,7)(9,11,13,12)(10,15,14,16)(17,20,22,27)(18,23,19,25)(21,29,30,28)(24,32,26,31)(33,36,37,44)(34,39,40,46)(35,42,43,48)(38,45,41,47);(1,4,5,2)(3,8,7,6)(9,12,13,11)(10,16,14,15)(17,21)(18,24)(19,26)(20,28)(22,30)(23,31)(25,32)(27,29)(33,34)(35,38)(36,39)(37,40)(41,43)(42,45)(44,46)(47,48)Polycyclic presentation:
2^9|a^2=fg,b^2=fi,c^2=fg,d^2=gh,e^2=hi,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, 2709072);