Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #6132643 |
| 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 | |
1396 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)(3,7)(4,8)(5,6)(9,10,13,12)(11,14,16,15)(19,25)(20,27)(22,28)(23,30)(33,34)(35,41)(36,40)(37,38)(39,43)(42,47)(44,45)(46,48);(1,2,5,6)(3,4,8,7)(9,10)(11,15)(12,13)(14,16)(19,25)(22,28)(26,31)(29,32)(34,38)(36,44)(39,41)(42,48);(2,6)(4,7)(9,11,13,16)(10,14,12,15)(17,18,21,24)(19,22,25,28)(20,23,27,30)(26,29,31,32)(33,35,37,43)(34,39,38,41)(36,42,44,48)(40,46,45,47);(1,3)(2,7)(4,6)(5,8)(9,12,13,10)(11,14,16,15)(17,19)(18,22)(20,26)(21,25)(23,29)(24,28)(27,31)(30,32)(33,36,37,44)(34,40,38,45)(35,42,43,48)(39,46,41,47);(1,4,5,7)(2,3,6,8)(9,11)(10,15)(12,14)(13,16)(17,20)(18,23)(19,26)(21,27)(22,29)(24,30)(25,31)(28,32)(33,34)(35,41)(36,40)(37,38)(39,43)(42,47)(44,45)(46,48)Polycyclic presentation:
2^9|a^2=fg,b^2=fgh,c^2=ghi,d^2=gh,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dfi,e^b=eg,d^c=dfg,e^c=ef62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6159866);