Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3069037 |
| 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 | |
1322 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 48 points:
(11,15)(13,16)(17,18)(19,22)(20,23)(21,24)(25,27)(26,29)(28,31)(30,32)(33,34,38,39)(35,36,40,44)(37,45,46,41)(42,47,48,43);(2,5)(6,8)(9,10,12,14)(11,13,15,16)(18,19)(20,27)(21,29)(31,32)(34,39)(35,40)(37,46)(43,47);(1,2,4,5)(3,6,7,8)(17,19,22,18)(20,23,27,25)(21,26,29,24)(28,31,30,32)(33,35)(34,36)(37,42)(38,40)(39,44)(41,43)(45,47)(46,48);(9,11)(10,13)(12,15)(14,16)(17,20,22,27)(18,23,19,25)(21,30,29,28)(24,32,26,31)(33,36)(34,40)(35,39)(37,47)(38,44)(41,42)(43,46)(45,48);(1,3,4,7)(2,6,5,8)(17,21)(18,24)(19,26)(20,28)(22,29)(23,31)(25,32)(27,30)(33,37,38,46)(34,41,39,45)(35,42,40,48)(36,43,44,47)Polycyclic presentation:
2^9|a^2=fg,b^2=i,c^2=fgh,d^2=g,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3004302);