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, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1450 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(3,7)(6,8)(11,12)(15,16)(17,18)(19,22)(20,25)(21,26)(23,27)(24,29)(28,30)(31,32)(33,34,37,40)(35,41,42,38)(36,39,44,46)(43,47,48,45);(2,5)(6,8)(10,14)(15,16)(18,19)(20,27)(21,29)(30,31)(35,42)(38,41)(43,48)(45,47);(1,2,4,5)(3,6,7,8)(9,10)(11,15)(12,16)(13,14)(17,19,22,18)(20,23,27,25)(21,26,29,24)(28,30,32,31)(33,34)(35,38)(36,39)(37,40)(41,42)(43,45)(44,46)(47,48);(9,11,13,12)(10,15,14,16)(17,20)(18,23)(19,25)(21,28)(22,27)(24,30)(26,31)(29,32)(33,35)(34,38)(36,43)(37,42)(39,45)(40,41)(44,48)(46,47);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,16,14,15)(17,21)(18,24)(19,26)(20,28)(22,29)(23,30)(25,31)(27,32)(33,36,37,44)(34,39,40,46)(35,43,42,48)(38,45,41,47)Polycyclic presentation:
2^9|a^2=fg,c^2=ghi,d^2=gi,e^2=gh,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, 4433501);