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 | |
1320 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(11,15)(13,16)(17,18,21,24)(19,25,26,22)(20,28,29,23)(27,30,32,31)(33,34)(35,39)(36,40)(37,38)(41,47)(42,46)(43,48)(44,45);(1,2,4,6)(3,5,7,8)(9,10,12,14)(11,13,15,16)(19,26)(22,25)(27,32)(30,31)(34,38)(39,46)(40,44)(47,48);(1,2)(3,5)(4,6)(7,8)(17,18)(19,22)(20,23)(21,24)(25,26)(27,30)(28,29)(31,32)(33,35,37,42)(34,39,38,46)(36,43,45,41)(40,48,44,47);(9,11)(10,13)(12,15)(14,16)(17,19,21,26)(18,22,24,25)(20,27,29,32)(23,30,28,31)(33,34)(35,39)(36,44)(37,38)(40,45)(41,48)(42,46)(43,47);(1,3,4,7)(2,5,6,8)(17,20,21,29)(18,23,24,28)(19,27,26,32)(22,30,25,31)(33,36)(34,40)(35,41)(37,45)(38,44)(39,47)(42,43)(46,48)Polycyclic presentation:
2^9|a^2=g,b^2=ghi,c^2=f,d^2=g,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, 3291981);