Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5035983 |
| 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:
(1,2,5,8)(3,6,4,7)(9,10,14,12)(11,15,13,16)(17,18,21,22)(19,25,26,23)(20,28,29,24)(27,30,32,31)(33,34)(35,39)(36,38)(37,44)(40,46)(41,43)(42,48)(45,47);(1,3,5,4)(2,6,8,7)(9,11,14,13)(10,15,12,16)(18,22)(19,26)(20,29)(30,31)(34,36)(35,41)(40,44)(42,45);(1,3,5,4)(2,6,8,7)(9,12,14,10)(11,15,13,16)(17,18)(19,23)(20,28)(21,22)(24,29)(25,26)(27,31)(30,32)(33,35,38,41)(34,39,36,43)(37,45,46,42)(40,48,44,47);(1,2)(3,6)(4,7)(5,8)(9,11,14,13)(10,16,12,15)(17,19,21,26)(18,23,22,25)(20,27,29,32)(24,30,28,31)(33,36,38,34)(35,39,41,43)(37,44,46,40)(42,47,45,48);(1,4,5,3)(2,7,8,6)(9,13,14,11)(10,15,12,16)(17,20)(18,24)(19,27)(21,29)(22,28)(23,30)(25,31)(26,32)(33,37,38,46)(34,40,36,44)(35,42,41,45)(39,47,43,48)Polycyclic presentation:
2^9|a^2=fgi,b^2=fi,c^2=gi,d^2=h,e^2=gi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5111006);