Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5605735 |
| 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 | |
1422 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)(3,7)(9,10,14,16)(11,12,13,15)(20,27)(24,29)(25,31)(30,32)(33,34,37,38)(35,41,43,39)(36,40,44,46)(42,48,45,47);(1,2)(3,6)(4,5)(7,8)(9,11,14,13)(10,15,16,12)(18,22)(23,28)(24,29)(30,32)(34,38)(36,44)(39,41)(42,45);(3,7)(6,8)(9,12,14,15)(10,13,16,11)(17,18,21,22)(19,23,26,28)(20,24,27,29)(25,30,31,32)(33,34)(35,39)(36,40)(37,38)(41,43)(42,47)(44,46)(45,48);(1,3,4,7)(2,6,5,8)(9,13,14,11)(10,15,16,12)(17,19,21,26)(18,23,22,28)(20,25,27,31)(24,30,29,32)(33,35,37,43)(34,39,38,41)(36,45,44,42)(40,48,46,47);(1,2)(3,8)(4,5)(6,7)(9,10)(11,15)(12,13)(14,16)(17,20)(18,24)(19,25)(21,27)(22,29)(23,30)(26,31)(28,32)(33,36,37,44)(34,40,38,46)(35,42,43,45)(39,47,41,48)Polycyclic presentation:
2^9|a^2=fgi,b^2=fi,c^2=fghi,d^2=fgh,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5700946);