Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4308740 |
| automorphism | Aut(G) order 2097152 |
| 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 | |
1402 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)(11,15)(13,16)(18,22)(19,25)(20,27)(30,32)(33,34,38,39)(35,41,42,36)(37,45,46,40)(43,44,48,47);(9,10,12,14)(11,13,15,16)(18,22)(23,28)(24,29)(30,32)(34,39)(35,42)(40,45)(43,48);(1,2,4,5)(3,6,7,8)(17,18,21,22)(19,23,25,28)(20,24,27,29)(26,30,31,32)(33,35,38,42)(34,36,39,41)(37,43,46,48)(40,44,45,47);(1,3)(2,6)(4,7)(5,8)(9,11)(10,13)(12,15)(14,16)(17,19,21,25)(18,23,22,28)(20,26,27,31)(24,30,29,32)(33,36)(34,35)(37,44)(38,41)(39,42)(40,43)(45,48)(46,47);(17,20)(18,24)(19,26)(21,27)(22,29)(23,30)(25,31)(28,32)(33,37,38,46)(34,40,39,45)(35,43,42,48)(36,44,41,47)Polycyclic presentation:
2^9|a^2=f,b^2=fgi,c^2=g,d^2=fh,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^c=df80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4216907);