Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4131229 |
| 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 | |
1364 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,8)(3,6,4,7)(9,10,13,12)(11,14,16,15)(17,18)(19,22)(20,23)(21,26)(24,29)(25,28)(27,32)(30,31)(33,34,38,39)(35,40,42,36)(37,41,47,48)(43,46,45,44);(1,3,5,4)(2,6,8,7)(9,10,13,12)(11,15,16,14)(18,19)(21,29)(23,25)(27,30)(34,39)(35,42)(41,48)(43,45);(9,11,13,16)(10,14,12,15)(17,19,22,18)(20,23,28,25)(21,24,29,26)(27,32,30,31)(33,35)(34,40)(36,39)(37,45)(38,42)(41,44)(43,47)(46,48);(1,2)(3,6)(4,7)(5,8)(9,10,13,12)(11,14,16,15)(17,20,22,28)(18,23,19,25)(21,30,29,27)(24,32,26,31)(33,36,38,40)(34,35,39,42)(37,46,47,44)(41,45,48,43);(1,4,5,3)(2,7,8,6)(9,12,13,10)(11,15,16,14)(17,21)(18,24)(19,26)(20,27)(22,29)(23,31)(25,32)(28,30)(33,37,38,47)(34,41,39,48)(35,43,42,45)(36,44,40,46)Polycyclic presentation:
2^9|a^2=ghi,b^2=fhi,c^2=fgh,d^2=h,e^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=eg,d^c=dg,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, 4207183);