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 | |
1306 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,4)(3,7,8,6)(9,10,13,15)(11,12,14,16)(17,18)(19,25)(20,24)(21,22)(23,26)(27,32)(28,30)(29,31)(33,34)(35,40)(36,43)(37,39)(38,45)(41,46)(42,48)(44,47);(9,11)(10,12)(13,14)(15,16)(18,22)(19,26)(24,28)(27,29)(33,35,39,36)(34,40,37,43)(38,42,46,44)(41,47,45,48);(1,2,5,4)(3,6,8,7)(9,12)(10,14)(11,15)(13,16)(17,18)(19,23)(20,28)(21,22)(24,30)(25,26)(27,32)(29,31)(33,36,39,35)(34,40,37,43)(38,44,46,42)(41,47,45,48);(1,3,5,8)(2,6,4,7)(17,19,21,26)(18,23,22,25)(20,29,30,27)(24,32,28,31)(33,37,39,34)(35,40,36,43)(38,45,46,41)(42,47,44,48);(1,4,5,2)(3,7,8,6)(17,20)(18,24)(19,27)(21,30)(22,28)(23,31)(25,32)(26,29)(33,38)(34,41)(35,42)(36,44)(37,45)(39,46)(40,47)(43,48)Polycyclic presentation:
2^9|a^2=ghi,b^2=g,c^2=fi,d^2=i,e^2=fgi,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, 3734109);