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)(17,18,22,19)(20,25,28,23)(21,26,29,24)(27,31,30,32)(33,34)(35,38)(36,44)(37,41)(39,43)(40,45)(42,47)(46,48);(10,13)(14,16)(18,19)(21,29)(23,25)(27,30)(33,35,37,43)(34,38,41,39)(36,42,45,48)(40,46,44,47);(1,2,5,4)(3,6,7,8)(9,10,12,13)(11,14,15,16)(17,19,22,18)(20,23,28,25)(21,26,29,24)(27,31,30,32)(33,35)(34,39)(36,42)(37,43)(38,41)(40,47)(44,46)(45,48);(1,3)(2,6)(4,8)(5,7)(17,20,22,28)(18,23,19,25)(21,30,29,27)(24,32,26,31)(33,34)(35,39)(36,40)(37,41)(38,43)(42,47)(44,45)(46,48);(1,4,5,2)(3,8,7,6)(9,11,12,15)(10,14,13,16)(17,21)(18,24)(19,26)(20,27)(22,29)(23,31)(25,32)(28,30)(33,36,37,45)(34,40,41,44)(35,42,43,48)(38,46,39,47)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=fghi,d^2=f,e^2=hi,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, 3162294);