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 | |
1374 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(9,10,12,14)(11,13,15,16)(17,18,22,19)(20,25,28,23)(21,26,29,24)(27,31,30,32)(33,34)(35,41)(36,40)(37,38)(39,42)(43,47)(44,45)(46,48);(2,5)(6,8)(9,11,12,15)(10,13,14,16)(18,19)(21,29)(23,25)(27,30)(34,38)(35,42)(36,44)(46,47);(1,2)(3,6)(4,5)(7,8)(9,11)(10,13)(12,15)(14,16)(17,19,22,18)(20,23,28,25)(21,26,29,24)(27,31,30,32)(33,35)(34,39)(36,43)(37,42)(38,41)(40,46)(44,48)(45,47);(1,3,4,7)(2,6,5,8)(9,10)(11,13)(12,14)(15,16)(17,20,22,28)(18,23,19,25)(21,30,29,27)(24,32,26,31)(33,34)(35,41)(36,45)(37,38)(39,42)(40,44)(43,46)(47,48);(17,21,22,29)(18,24,19,26)(20,27,28,30)(23,31,25,32)(33,36,37,44)(34,40,38,45)(35,43,42,48)(39,46,41,47)Polycyclic presentation:
2^9|a^2=i,b^2=fgi,c^2=fg,d^2=fhi,e^2=f,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, 3069607);