Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3069037 |
| 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 | |
1366 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)(17,18,21,22)(19,25,26,23)(20,24,28,30)(27,32,29,31)(33,34)(35,39)(36,44)(37,38)(40,46)(41,48)(42,43)(45,47);(9,10,13,12)(11,15,16,14)(18,22)(19,26)(20,28)(31,32)(34,37)(39,42)(40,44)(47,48);(1,2,5,4)(3,6,7,8)(9,11,13,16)(10,14,12,15)(17,18)(19,25)(20,24)(21,22)(23,26)(27,32)(28,30)(29,31)(33,35,38,43)(34,39,37,42)(36,45,46,41)(40,48,44,47);(1,3)(2,6)(4,8)(5,7)(9,10,13,12)(11,14,16,15)(17,19,21,26)(18,23,22,25)(20,29,28,27)(24,32,30,31)(33,36,38,46)(34,40,37,44)(35,41,43,45)(39,47,42,48);(1,4,5,2)(3,8,7,6)(9,12,13,10)(11,15,16,14)(17,20,21,28)(18,24,22,30)(19,27,26,29)(23,31,25,32)(33,37,38,34)(35,42,43,39)(36,40,46,44)(41,47,45,48)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=hi,d^2=fgh,e^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,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, 3173201);