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, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1381 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,3)(4,7,8,6)(9,10,13,11)(12,15,16,14)(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);(1,2,5,3)(4,7,8,6)(9,11,13,10)(12,15,16,14)(18,22)(19,26)(20,29)(30,31)(34,36)(35,41)(40,44)(42,45);(2,3)(6,7)(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);(1,3,5,2)(4,7,8,6)(9,12,13,16)(10,14,11,15)(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);(1,4,5,8)(2,6,3,7)(17,20)(18,24)(19,27)(21,29)(22,28)(23,30)(25,31)(26,32)(33,37)(34,40)(35,42)(36,44)(38,46)(39,47)(41,45)(43,48)Polycyclic presentation:
2^9|a^2=fgi,b^2=hi,c^2=gh,d^2=fhi,e^2=fh,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, 5030244);