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)(17,18,22,23)(19,26,24,20)(21,29,32,25)(27,28,30,31)(33,34)(35,41)(36,39)(37,40)(38,43)(42,48)(44,46)(45,47);(9,11)(10,12)(13,14)(15,16)(18,23)(19,24)(25,29)(27,30)(33,34,37,40)(35,41,43,38)(36,39,46,44)(42,48,45,47);(1,2,5,4)(3,6,8,7)(9,12)(10,14)(11,15)(13,16)(17,19,22,24)(18,20,23,26)(21,30,32,27)(25,31,29,28)(33,34)(35,38)(36,44)(37,40)(39,46)(41,43)(42,48)(45,47);(1,3,5,8)(2,6,4,7)(17,20)(18,24)(19,23)(21,31)(22,26)(25,27)(28,32)(29,30)(33,35)(34,38)(36,45)(37,43)(39,48)(40,41)(42,46)(44,47);(1,4,5,2)(3,7,8,6)(17,21)(18,25)(19,27)(20,28)(22,32)(23,29)(24,30)(26,31)(33,36)(34,39)(35,42)(37,46)(38,47)(40,44)(41,48)(43,45)Polycyclic presentation:
2^9|a^2=ghi,b^2=fg,c^2=fgi,d^2=fi,e^2=fi,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, 3733735);