Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5473759 |
| 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 | |
1356 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,5)(6,8)(10,14)(11,16)(17,18,22,20)(19,25,27,23)(21,24,29,28)(26,31,32,30)(33,34)(35,38)(36,44)(37,41)(39,43)(40,46)(42,48)(45,47);(9,10)(11,15)(12,16)(13,14)(21,29)(24,28)(26,32)(30,31)(33,35,37,43)(34,38,41,39)(36,45,46,42)(40,48,44,47);(10,14)(12,15)(17,19,22,27)(18,23,20,25)(21,26,29,32)(24,30,28,31)(33,35)(34,39)(36,42)(37,43)(38,41)(40,48)(44,47)(45,46);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,14)(13,16)(17,20,22,18)(19,23,27,25)(21,28,29,24)(26,30,32,31)(33,34)(35,38)(36,40)(37,41)(39,43)(42,47)(44,46)(45,48);(1,3,4,7)(2,6,5,8)(9,12,13,15)(10,11,14,16)(17,21,22,29)(18,24,20,28)(19,26,27,32)(23,30,25,31)(33,36)(34,40)(35,42)(37,46)(38,47)(39,48)(41,44)(43,45)Polycyclic presentation:
2^9|a^2=fgh,b^2=fh,c^2=fgh,d^2=fgh,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=efg,d^c=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5480098);