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 | |
1246 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)(5,8)(10,14)(11,17)(12,19)(22,24)(25,26,30,27)(28,33,36,31)(29,34,37,32)(35,38,40,39)(41,42)(43,49)(44,52)(45,46)(47,50)(48,53)(51,54)(55,56);(1,2,4,6)(3,5,7,8)(10,14)(11,17)(16,21)(18,23)(26,27)(29,37)(31,33)(35,40)(42,46)(43,50)(44,53)(54,55);(9,10)(11,15)(12,16)(13,14)(17,20)(18,22)(19,21)(23,24)(25,27,30,26)(28,31,36,33)(29,34,37,32)(35,38,40,39)(41,43)(42,47)(44,51)(45,50)(46,49)(48,54)(52,55)(53,56);(1,3)(2,5)(4,7)(6,8)(9,11,13,17)(10,15,14,20)(12,18,19,23)(16,22,21,24)(25,28,30,36)(26,31,27,33)(29,35,37,40)(32,38,34,39)(41,42)(43,49)(44,48)(45,46)(47,50)(51,55)(52,53)(54,56);(9,12)(10,16)(11,18)(13,19)(14,21)(15,22)(17,23)(20,24)(25,29)(26,32)(27,34)(28,35)(30,37)(31,38)(33,39)(36,40)(41,44,45,53)(42,48,46,52)(43,51,50,56)(47,54,49,55)Polycyclic presentation:
2^9|a^2=gh,b^2=fgi,c^2=gh,d^2=fg,e^2=fgh,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, 4389718);