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 | |
1394 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)(4,5)(6,8)(9,10,12,14)(11,15,16,13)(19,25)(23,28)(26,31)(30,32)(33,34)(35,40)(36,37)(38,41)(39,42)(43,47)(44,45)(46,48);(2,4)(6,7)(11,16)(13,15)(18,22)(19,25)(24,29)(26,31)(33,35,39,36)(34,40,42,37)(38,44,46,43)(41,45,48,47);(2,4)(6,7)(17,18)(19,23)(20,24)(21,22)(25,28)(26,30)(27,29)(31,32)(33,36,39,35)(34,40,42,37)(38,44,46,43)(41,47,48,45);(1,3,5,8)(2,6,4,7)(9,11)(10,13)(12,16)(14,15)(17,19)(18,23)(20,26)(21,25)(22,28)(24,30)(27,31)(29,32)(33,37,39,40)(34,36,42,35)(38,45,46,47)(41,44,48,43);(1,4,5,2)(3,7,8,6)(9,10)(11,13)(12,14)(15,16)(17,20,21,27)(18,24,22,29)(19,26,25,31)(23,30,28,32)(33,38,39,46)(34,41,42,48)(35,43,36,44)(37,45,40,47)Polycyclic presentation:
2^9|a^2=fhi,b^2=fg,c^2=fg,d^2=g,e^2=gh,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, 6085632);