Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| 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 | |
1356 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)(11,13)(12,14)(15,16)(17,18,22,25)(19,21,26,23)(20,24,30,32)(27,29,28,31)(34,35)(37,45)(39,41)(43,48);(4,8)(6,7)(11,15)(13,16)(19,26)(21,23)(27,28)(29,31)(33,34,38,35)(36,41,44,39)(37,42,45,40)(43,46,48,47);(2,3)(6,7)(9,10,12,14)(11,13,15,16)(17,19)(18,23)(20,28)(21,25)(22,26)(24,29)(27,30)(31,32)(33,35,38,34)(36,41,44,39)(37,40,45,42)(43,46,48,47);(1,3,5,2)(4,7,8,6)(9,10)(11,13)(12,14)(15,16)(17,20)(18,24)(19,27)(21,29)(22,30)(23,31)(25,32)(26,28)(33,36)(34,39)(35,41)(37,43)(38,44)(40,46)(42,47)(45,48);(1,4)(2,6)(3,7)(5,8)(9,11)(10,13)(12,15)(14,16)(17,21,22,23)(18,19,25,26)(20,29,30,31)(24,27,32,28)(33,37,38,45)(34,40,35,42)(36,43,44,48)(39,46,41,47)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=ghi,d^2=i,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5238765);