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 | |
1470 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,4)(3,7,8,6)(18,22)(20,27)(23,28)(26,31)(33,34)(35,39)(36,40)(37,38)(41,47)(42,46)(43,48)(44,45);(9,10,13,12)(11,15,16,14)(18,22)(19,25)(24,29)(26,31)(34,38)(39,46)(40,44)(47,48);(1,2,5,4)(3,6,8,7)(9,11,13,16)(10,14,12,15)(17,18,21,22)(19,23,25,28)(20,24,27,29)(26,30,31,32)(33,35,37,42)(34,39,38,46)(36,43,45,41)(40,48,44,47);(1,3,5,8)(2,6,4,7)(9,10,13,12)(11,14,16,15)(17,19)(18,23)(20,26)(21,25)(22,28)(24,30)(27,31)(29,32)(33,34)(35,39)(36,44)(37,38)(40,45)(41,48)(42,46)(43,47);(1,4,5,2)(3,7,8,6)(9,12,13,10)(11,15,16,14)(17,20)(18,24)(19,26)(21,27)(22,29)(23,30)(25,31)(28,32)(33,36,37,45)(34,40,38,44)(35,41,42,43)(39,47,46,48)Polycyclic presentation:
2^9|a^2=i,b^2=gh,c^2=fhi,d^2=ghi,e^2=fghi,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, 3764029);