Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| 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 | |
1368 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)(17,18,21,24)(19,22,25,29)(20,27,28,23)(26,32,31,30)(33,34,38,39)(35,42,40,36)(37,45,48,41)(43,44,46,47);(2,5)(6,8)(9,10,13,12)(11,15,16,14)(19,25)(22,29)(26,31)(30,32)(34,39)(35,40)(41,45)(43,46);(1,2,4,5)(3,6,7,8)(9,11,13,16)(10,14,12,15)(17,18)(19,22)(20,23)(21,24)(25,29)(26,30)(27,28)(31,32)(33,35,38,40)(34,36,39,42)(37,46,48,43)(41,47,45,44);(1,3,4,7)(2,6,5,8)(9,12,13,10)(11,15,16,14)(17,19)(18,22)(20,26)(21,25)(23,30)(24,29)(27,32)(28,31)(33,36)(34,40)(35,39)(37,47)(38,42)(41,43)(44,48)(45,46);(17,20,21,28)(18,23,24,27)(19,26,25,31)(22,30,29,32)(33,37)(34,41)(35,43)(36,44)(38,48)(39,45)(40,46)(42,47)Polycyclic presentation:
2^9|a^2=g,b^2=hi,c^2=h,d^2=fh,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,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, 3287585);