Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4465744 |
| 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 | |
1354 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,3)(4,7,8,6)(9,10,13,15)(11,12,16,14)(18,22)(23,28)(24,29)(30,32)(33,34,37,40)(35,41,42,38)(36,39,44,46)(43,47,48,45);(1,2)(3,5)(4,7)(6,8)(9,11,13,16)(10,14,15,12)(19,25)(23,28)(26,31)(30,32)(35,42)(36,44)(38,41)(39,46);(2,3)(6,7)(10,15)(11,16)(17,18,21,22)(19,23,25,28)(20,24,27,29)(26,30,31,32)(33,35,37,42)(34,38,40,41)(36,43,44,48)(39,45,46,47);(1,3,5,2)(4,7,8,6)(9,11)(10,14)(12,15)(13,16)(17,19)(18,23)(20,26)(21,25)(22,28)(24,30)(27,31)(29,32)(33,36)(34,39)(35,43)(37,44)(38,45)(40,46)(41,47)(42,48);(1,4,5,8)(2,6,3,7)(9,12,13,14)(10,11,15,16)(17,20,21,27)(18,24,22,29)(19,26,25,31)(23,30,28,32)(33,34)(35,38)(36,39)(37,40)(41,42)(43,45)(44,46)(47,48)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=fg,d^2=fi,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^b=ef,d^c=df68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4419480);