Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2760365 |
| 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 | |
1366 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(3,7)(6,8)(17,18,21,24)(19,22,25,29)(20,27,28,23)(26,32,31,30)(33,34,38,36)(35,39,42,43)(37,44,45,40)(41,47,48,46);(1,2,5,4)(3,6,7,8)(9,10,13,12)(11,15,16,14)(19,25)(22,29)(26,31)(30,32)(34,36)(39,43)(40,44)(46,47);(9,11,13,16)(10,14,12,15)(17,18)(19,22)(20,23)(21,24)(25,29)(26,30)(27,28)(31,32)(33,35,38,42)(34,39,36,43)(37,41,45,48)(40,46,44,47);(1,3)(2,6)(4,8)(5,7)(17,19,21,25)(18,22,24,29)(20,26,28,31)(23,30,27,32)(33,36,38,34)(35,39,42,43)(37,40,45,44)(41,47,48,46);(1,4,5,2)(3,8,7,6)(9,12,13,10)(11,15,16,14)(17,20,21,28)(18,23,24,27)(19,26,25,31)(22,30,29,32)(33,37,38,45)(34,40,36,44)(35,41,42,48)(39,46,43,47)Polycyclic presentation:
2^9|a^2=fg,b^2=hi,c^2=fh,d^2=fg,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2708454);