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 | |
1418 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)(3,6)(4,8)(5,7)(10,14)(12,16)(17,18,22,19)(20,23,28,25)(21,26,29,24)(27,31,32,30)(33,34)(35,41)(36,40)(37,38)(39,42)(43,48)(44,46)(45,47);(1,3,5,8)(2,6,7,4)(10,14)(11,15)(18,19)(21,29)(23,25)(27,32)(34,37)(35,42)(36,44)(47,48);(1,3)(2,4)(5,8)(6,7)(9,10)(11,12)(13,14)(15,16)(17,19,22,18)(20,23,28,25)(21,26,29,24)(27,30,32,31)(33,35)(34,39)(36,45)(37,41)(38,42)(40,48)(43,44)(46,47);(1,3)(2,4)(5,8)(6,7)(9,11)(10,12)(13,15)(14,16)(17,20,22,28)(18,23,19,25)(21,27,29,32)(24,30,26,31)(33,36,38,44)(34,40,37,46)(35,43,42,45)(39,47,41,48);(1,4)(2,3)(5,6)(7,8)(9,12)(10,11)(13,16)(14,15)(17,21)(18,24)(19,26)(20,27)(22,29)(23,30)(25,31)(28,32)(33,37,38,34)(35,41,42,39)(36,46,44,40)(43,48,45,47)Polycyclic presentation:
2^9|a^2=gh,b^2=i,c^2=gh,d^2=f,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, 4389396);