Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3069037 |
| 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 | |
1392 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)(11,15)(13,16)(17,18,22,19)(20,23,28,25)(21,26,29,24)(27,32,30,31)(33,34,38,36)(35,41,43,39)(37,40,46,44)(42,48,45,47);(2,5)(6,8)(18,19)(21,29)(23,25)(27,30)(33,35,38,43)(34,39,36,41)(37,45,46,42)(40,48,44,47);(1,2)(3,6)(4,5)(7,8)(9,10,12,14)(11,13,15,16)(17,19,22,18)(20,23,28,25)(21,26,29,24)(27,31,30,32)(33,35)(34,39)(36,41)(37,42)(38,43)(40,47)(44,48)(45,46);(1,3)(2,6)(4,7)(5,8)(9,11,12,15)(10,13,14,16)(17,20,22,28)(18,23,19,25)(21,30,29,27)(24,32,26,31)(33,36,38,34)(35,39,43,41)(37,40,46,44)(42,48,45,47);(17,21,22,29)(18,24,19,26)(20,27,28,30)(23,31,25,32)(33,37,38,46)(34,40,36,44)(35,42,43,45)(39,47,41,48)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=ghi,d^2=hi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3082042);