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, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1464 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(3,7)(6,8)(17,18)(19,22)(20,25)(21,26)(23,28)(24,30)(27,31)(29,32)(33,34,38,36)(35,39,43,41)(37,40,46,44)(42,47,45,48);(9,10,13,12)(11,15,16,14)(18,19)(23,25)(24,26)(31,32)(34,36)(39,41)(40,44)(47,48);(1,2,5,4)(3,6,7,8)(9,11,13,16)(10,14,12,15)(17,19,22,18)(20,25,28,23)(21,24,30,26)(27,31,29,32)(33,35,38,43)(34,39,36,41)(37,45,46,42)(40,48,44,47);(1,3)(2,6)(4,8)(5,7)(17,20,22,28)(18,23,19,25)(21,29,30,27)(24,32,26,31)(33,36,38,34)(35,41,43,39)(37,40,46,44)(42,47,45,48);(1,4,5,2)(3,8,7,6)(9,12,13,10)(11,15,16,14)(17,21,22,30)(18,24,19,26)(20,27,28,29)(23,31,25,32)(33,37)(34,40)(35,42)(36,44)(38,46)(39,47)(41,48)(43,45)Polycyclic presentation:
2^9|a^2=fg,b^2=gh,c^2=fhi,d^2=f,e^2=ghi,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, 3591766);