Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4465744 |
| 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 | |
1238 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 56 points:
(3,7)(6,8)(9,10,13,14)(11,17,18,15)(12,20,21,16)(19,22,24,23)(25,26)(27,33)(28,36)(29,30)(31,35)(32,37)(34,38)(39,40)(42,43)(44,51)(48,50)(52,56);(10,14)(11,18)(12,21)(22,23)(26,30)(28,37)(31,33)(34,40)(41,42,46,43)(44,49,51,47)(45,50,53,48)(52,54,56,55);(1,2,5,4)(3,6,7,8)(9,10)(11,17)(12,16)(13,14)(15,18)(19,23)(20,21)(22,24)(25,26)(27,33)(28,32)(29,30)(31,35)(34,39)(36,37)(38,40)(41,43,46,42)(44,47,51,49)(45,50,53,48)(52,54,56,55);(1,3)(2,6)(4,8)(5,7)(9,11)(10,15)(12,19)(13,18)(14,17)(16,22)(20,23)(21,24)(25,27,29,35)(26,31,30,33)(28,34,37,40)(32,38,36,39)(41,44,46,51)(42,47,43,49)(45,52,53,56)(48,54,50,55);(1,4,5,2)(3,8,7,6)(9,12,13,21)(10,16,14,20)(11,19,18,24)(15,22,17,23)(25,28)(26,32)(27,34)(29,37)(30,36)(31,38)(33,39)(35,40)(41,45)(42,48)(43,50)(44,52)(46,53)(47,54)(49,55)(51,56)Polycyclic presentation:
2^9|a^2=fgh,b^2=fh,c^2=hi,d^2=gh,e^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^b=ef,d^c=df68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4504831);