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 | |
1300 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,7)(3,8,4,6)(11,15)(13,16)(17,18,21,22)(19,25,26,23)(20,28,29,24)(27,30,32,31)(33,34)(35,36)(37,41)(38,39)(40,43)(42,44)(45,46)(47,48);(1,3,5,4)(2,6,7,8)(18,22)(19,26)(20,29)(30,31)(34,39)(36,40)(37,45)(42,48);(3,4)(6,8)(9,10,12,14)(11,13,15,16)(17,18)(19,25)(20,24)(21,22)(23,26)(27,31)(28,29)(30,32)(33,35,38,43)(34,40,39,36)(37,42,45,48)(41,47,46,44);(1,4,5,3)(2,6,7,8)(9,11,12,15)(10,13,14,16)(17,19)(18,23)(20,27)(21,26)(22,25)(24,30)(28,31)(29,32)(33,36,38,40)(34,35,39,43)(37,44,45,47)(41,42,46,48);(1,2)(3,8)(4,6)(5,7)(17,20,21,29)(18,24,22,28)(19,27,26,32)(23,30,25,31)(33,37)(34,41)(35,42)(36,44)(38,45)(39,46)(40,47)(43,48)Polycyclic presentation:
2^9|a^2=h,b^2=fg,c^2=gi,d^2=fi,e^2=fgh,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, 4431854);