Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| 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 | |
1300 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(12,16)(14,15)(17,18)(19,23)(20,24)(21,22)(25,30)(26,31)(27,29)(28,32)(33,34,38,41)(35,42,39,36)(37,45,46,40)(43,44,47,48);(2,5)(6,8)(9,10,13,11)(12,14,16,15)(18,22)(19,25)(24,27)(26,28)(35,39)(36,42)(43,47)(44,48);(1,2)(3,6)(4,5)(7,8)(9,11,13,10)(12,15,16,14)(17,18)(19,23)(20,27)(21,22)(24,29)(25,30)(26,32)(28,31)(33,35,38,39)(34,36,41,42)(37,43,46,47)(40,44,45,48);(9,12,13,16)(10,14,11,15)(17,19,21,25)(18,23,22,30)(20,28,29,26)(24,32,27,31)(33,36)(34,39)(35,41)(37,44)(38,42)(40,47)(43,45)(46,48);(1,3,4,7)(2,6,5,8)(17,20,21,29)(18,24,22,27)(19,26,25,28)(23,31,30,32)(33,37)(34,40)(35,43)(36,44)(38,46)(39,47)(41,45)(42,48)Polycyclic presentation:
2^9|a^2=g,b^2=gi,c^2=i,d^2=fgi,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3479249);