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 | |
1326 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,6)(3,7,8,5)(10,11)(14,15)(18,22)(19,25)(20,28)(30,32)(33,34)(35,40)(36,41)(37,42)(38,39)(43,48)(44,45)(46,47);(3,8)(5,7)(9,10,13,11)(12,14,16,15)(19,25)(20,28)(23,26)(24,29)(34,39)(36,40)(42,47)(44,48);(10,11)(12,16)(17,18,21,22)(19,26,25,23)(20,24,28,29)(27,32,31,30)(33,35)(34,40)(36,39)(37,45)(38,41)(42,44)(43,46)(47,48);(1,3)(2,5)(4,8)(6,7)(9,11,13,10)(12,15,16,14)(17,19,21,25)(18,23,22,26)(20,27,28,31)(24,30,29,32)(33,36,38,40)(34,41,39,35)(37,44,46,48)(42,45,47,43);(1,2)(3,5)(4,6)(7,8)(9,12,13,16)(10,14,11,15)(17,20)(18,24)(19,27)(21,28)(22,29)(23,30)(25,31)(26,32)(33,37)(34,42)(35,43)(36,44)(38,46)(39,47)(40,48)(41,45)Polycyclic presentation:
2^9|a^2=ghi,b^2=h,c^2=g,d^2=fg,e^2=h,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, 6086901);