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 | |
1316 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)(6,8)(11,12)(15,16)(17,18,21,22)(19,23,25,30)(20,27,29,24)(26,32,28,31)(33,34)(35,37)(36,40)(38,39)(41,46)(42,45)(43,48)(44,47);(1,2,5,4)(3,6,7,8)(10,14)(15,16)(18,22)(19,25)(24,27)(26,28)(34,39)(35,41)(40,47)(42,48);(9,10)(11,15)(12,16)(13,14)(17,18)(19,23)(20,24)(21,22)(25,30)(26,31)(27,29)(28,32)(33,35,38,41)(34,37,39,46)(36,42,44,48)(40,45,47,43);(1,3)(2,6)(4,8)(5,7)(9,11,13,12)(10,15,14,16)(17,19)(18,23)(20,28)(21,25)(22,30)(24,32)(26,29)(27,31)(33,36)(34,40)(35,42)(37,45)(38,44)(39,47)(41,48)(43,46);(1,4,5,2)(3,8,7,6)(9,12,13,11)(10,16,14,15)(17,20)(18,24)(19,26)(21,29)(22,27)(23,31)(25,28)(30,32)(33,37)(34,41)(35,39)(36,43)(38,46)(40,42)(44,45)(47,48)Polycyclic presentation:
2^9|a^2=g,b^2=fhi,c^2=fg,d^2=fh,e^2=i,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, 4826110);