Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4861280 |
| 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 | |
1298 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)(9,10,12,13)(11,15,16,14)(17,18)(19,24)(20,25)(21,29)(22,23)(26,31)(27,28)(30,32)(33,34,38,36)(35,39,43,41)(37,44,45,40)(42,47,48,46);(10,13)(14,15)(18,23)(19,25)(26,29)(27,32)(33,35,38,43)(34,39,36,41)(37,42,45,48)(40,46,44,47);(9,10)(11,14)(12,13)(15,16)(17,19,22,25)(18,24,23,20)(21,27,31,32)(26,30,29,28)(33,35)(34,39)(36,41)(37,42)(38,43)(40,46)(44,47)(45,48);(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,13)(14,16)(17,20,22,24)(18,25,23,19)(21,30,31,28)(26,27,29,32)(33,36,38,34)(35,41,43,39)(37,44,45,40)(42,47,48,46);(1,3,4,7)(2,6,5,8)(9,11,12,16)(10,14,13,15)(17,21,22,31)(18,26,23,29)(19,27,25,32)(20,28,24,30)(33,37,38,45)(34,40,36,44)(35,42,43,48)(39,46,41,47)Polycyclic presentation:
2^9|a^2=fgh,b^2=fh,c^2=fg,d^2=gh,e^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4819203);