Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4861280 |
| 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 | |
1320 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)(11,15)(13,16)(17,18,22,19)(20,25,28,23)(21,26,30,24)(27,31,29,32)(33,34)(35,41)(36,38)(37,40)(39,43)(42,47)(44,45)(46,48);(1,2,5,4)(3,6,7,8)(9,10,12,14)(11,13,15,16)(18,19)(23,25)(24,26)(31,32)(34,36)(39,41)(40,44)(46,47);(9,10)(11,13)(12,14)(15,16)(17,19,22,18)(20,25,28,23)(21,26,30,24)(27,32,29,31)(33,35)(34,39)(36,41)(37,42)(38,43)(40,46)(44,47)(45,48);(1,3)(2,6)(4,8)(5,7)(17,20)(18,23)(19,25)(21,29)(22,28)(24,32)(26,31)(27,30)(33,36,38,34)(35,41,43,39)(37,40,45,44)(42,46,48,47);(1,4,5,2)(3,8,7,6)(9,11)(10,13)(12,15)(14,16)(17,21)(18,24)(19,26)(20,27)(22,30)(23,31)(25,32)(28,29)(33,37,38,45)(34,40,36,44)(35,42,43,48)(39,46,41,47)Polycyclic presentation:
2^9|a^2=fg,b^2=ghi,c^2=fg,d^2=g,e^2=fgi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4822222);