Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4932388 |
| 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 | |
1362 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)(3,5)(4,6)(7,8)(9,10)(11,12)(13,16)(14,15)(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,29)(31,32)(33,34,37,40)(35,41,42,38)(36,44,45,39)(43,46,48,47);(9,11,14,13)(10,12,15,16)(18,22)(20,28)(23,25)(26,32)(35,42)(38,41)(43,48)(46,47);(2,3)(6,7)(10,15)(12,16)(17,18)(19,25)(20,24)(21,22)(23,27)(26,31)(28,29)(30,32)(33,35,37,42)(34,38,40,41)(36,43,45,48)(39,46,44,47);(1,3,5,2)(4,7,8,6)(9,12,14,16)(10,11,15,13)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,32)(29,31)(33,34)(35,41)(36,39)(37,40)(38,42)(43,47)(44,45)(46,48);(1,4,5,8)(2,6,3,7)(9,13,14,11)(10,16,15,12)(17,20,21,28)(18,24,22,29)(19,26,27,32)(23,30,25,31)(33,36,37,45)(34,39,40,44)(35,43,42,48)(38,46,41,47)Polycyclic presentation:
2^9|a^2=fh,b^2=gi,c^2=fh,d^2=fgh,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=ef,d^c=dfg80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4861777);