Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4131229 |
| 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 | |
1380 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,6)(3,5,7,8)(17,18)(19,22)(20,25)(21,26)(23,28)(24,29)(27,31)(30,32)(33,34)(35,41)(36,39)(37,40)(38,43)(42,48)(44,46)(45,47);(10,13)(14,16)(18,19)(21,29)(23,25)(27,30)(33,34,37,40)(35,38,43,41)(36,39,46,44)(42,47,45,48);(9,10)(11,14)(12,13)(15,16)(17,19,22,18)(20,23,28,25)(21,24,29,26)(27,32,30,31)(33,34)(35,38)(36,44)(37,40)(39,46)(41,43)(42,48)(45,47);(1,2)(3,5)(4,6)(7,8)(17,20,22,28)(18,23,19,25)(21,30,29,27)(24,32,26,31)(33,35)(34,38)(36,45)(37,43)(39,48)(40,41)(42,46)(44,47);(1,3,4,7)(2,5,6,8)(9,11,12,15)(10,14,13,16)(17,21,22,29)(18,24,19,26)(20,27,28,30)(23,31,25,32)(33,36,37,46)(34,39,40,44)(35,42,43,45)(38,47,41,48)Polycyclic presentation:
2^9|a^2=fi,b^2=fg,c^2=g,d^2=g,e^2=fhi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=eg,d^c=dg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4185032);