Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| 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 | |
1304 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,5)(6,8)(10,13)(14,15)(17,18,21,22)(19,25,27,23)(20,28,30,24)(26,31,29,32)(33,34)(35,39)(36,38)(37,44)(40,43)(41,45)(42,47)(46,48);(9,10)(11,14)(12,13)(15,16)(18,22)(23,25)(24,28)(31,32)(33,35,38,43)(34,39,36,40)(37,42,45,48)(41,46,44,47);(9,11,12,16)(10,14,13,15)(17,19)(18,23)(20,29)(21,27)(22,25)(24,32)(26,30)(28,31)(33,35)(34,40)(36,39)(37,42)(38,43)(41,47)(44,46)(45,48);(1,2)(3,6)(4,5)(7,8)(17,18)(19,23)(20,24)(21,22)(25,27)(26,31)(28,30)(29,32)(33,36,38,34)(35,39,43,40)(37,44,45,41)(42,46,48,47);(1,3,4,7)(2,6,5,8)(9,10)(11,15)(12,13)(14,16)(17,20)(18,24)(19,26)(21,30)(22,28)(23,31)(25,32)(27,29)(33,37,38,45)(34,41,36,44)(35,42,43,48)(39,46,40,47)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=fgh,d^2=h,e^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4669824);