Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| 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 | |
1378 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,5)(6,8)(17,18,21,22)(19,23,26,30)(20,24,29,28)(25,31,27,32)(33,34)(35,39)(36,40)(37,43)(38,41)(42,47)(44,45)(46,48);(10,13)(14,16)(18,22)(23,30)(24,28)(31,32)(33,35,38,43)(34,39,41,37)(36,44,46,42)(40,45,48,47);(9,10,12,13)(11,14,15,16)(17,19,21,26)(18,23,22,30)(20,27,29,25)(24,32,28,31)(33,35)(34,37)(36,42)(38,43)(39,41)(40,45)(44,46)(47,48);(1,2,4,5)(3,6,7,8)(17,18)(19,23)(20,28)(21,22)(24,29)(25,32)(26,30)(27,31)(33,36,38,46)(34,40,41,48)(35,42,43,44)(37,45,39,47);(1,3,4,7)(2,6,5,8)(9,11,12,15)(10,14,13,16)(17,20)(18,24)(19,25)(21,29)(22,28)(23,31)(26,27)(30,32)(33,37)(34,35)(36,45)(38,39)(40,42)(41,43)(44,48)(46,47)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=fgh,d^2=fgi,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3655622);