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) = C25×C4 |
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 | |
1352 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,6)(3,8,7,4)(9,10)(11,13)(12,14)(15,16)(18,22)(20,28)(23,25)(26,32)(33,34)(35,41)(36,40)(37,38)(39,43)(42,47)(44,45)(46,48);(3,7)(4,8)(9,11,14,15)(10,13,12,16)(18,22)(20,28)(23,25)(26,32)(34,38)(39,41)(40,44)(46,47);(2,6)(3,7)(10,12)(11,15)(17,18,21,22)(19,25,27,23)(20,24,28,29)(26,31,32,30)(33,34)(35,41)(36,44)(37,38)(39,43)(40,45)(42,46)(47,48);(1,3,5,7)(2,4,6,8)(9,12,14,10)(11,13,15,16)(17,19,21,27)(18,23,22,25)(20,26,28,32)(24,30,29,31)(33,35,37,43)(34,39,38,41)(36,42,45,48)(40,46,44,47);(1,4)(2,7)(3,6)(5,8)(9,13)(10,11)(12,15)(14,16)(17,20)(18,24)(19,26)(21,28)(22,29)(23,30)(25,31)(27,32)(33,36,37,45)(34,40,38,44)(35,42,43,48)(39,46,41,47)Polycyclic presentation:
2^9|a^2=fgh,b^2=ghi,c^2=fgi,d^2=fg,e^2=f,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, 6176664);