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 | |
1362 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)(11,15)(14,16)(17,18,22,23)(19,24,26,20)(21,25,30,31)(27,32,29,28)(33,34)(35,39)(36,42)(37,40)(38,41)(43,46)(44,48)(45,47);(10,13)(14,16)(18,23)(19,26)(25,31)(27,29)(33,35,38,36)(34,39,41,42)(37,43,45,44)(40,46,47,48);(9,10)(11,14)(12,13)(15,16)(17,19)(18,24)(20,23)(21,29)(22,26)(25,28)(27,30)(31,32)(33,36,38,35)(34,39,41,42)(37,44,45,43)(40,46,47,48);(1,2)(3,6)(4,5)(7,8)(17,20,22,24)(18,19,23,26)(21,28,30,32)(25,27,31,29)(33,34)(35,42)(36,39)(37,40)(38,41)(43,48)(44,46)(45,47);(1,3,4,7)(2,6,5,8)(9,11,12,15)(10,14,13,16)(17,21)(18,25)(19,27)(20,28)(22,30)(23,31)(24,32)(26,29)(33,37,38,45)(34,40,41,47)(35,43,36,44)(39,46,42,48)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=g,d^2=f,e^2=fghi,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, 4654517);