Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4465744 |
| 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 | |
1364 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)(5,8)(9,10)(11,15)(12,16)(13,14)(17,18,22,19)(20,23,28,25)(21,26,29,24)(27,31,32,30)(33,34)(35,40)(36,41)(37,42)(38,39)(43,48)(44,45)(46,47);(9,11,14,12)(10,15,13,16)(18,19)(21,29)(23,25)(27,32)(34,39)(37,40)(41,47)(45,48);(1,2,4,6)(3,5,7,8)(9,12,14,11)(10,15,13,16)(17,19,22,18)(20,23,28,25)(21,26,29,24)(27,30,32,31)(33,35,38,42)(34,40,39,37)(36,44,46,43)(41,45,47,48);(1,3,4,7)(2,5,6,8)(9,13,14,10)(11,15,12,16)(17,20,22,28)(18,23,19,25)(21,27,29,32)(24,30,26,31)(33,36,38,46)(34,41,39,47)(35,43,42,44)(37,45,40,48);(17,21)(18,24)(19,26)(20,27)(22,29)(23,30)(25,31)(28,32)(33,37,38,40)(34,42,39,35)(36,45,46,48)(41,44,47,43)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=fi,d^2=fi,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^b=ef,d^c=df68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4431560);