Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5473759 |
| 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 | |
1240 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,4,6)(3,5,7,8)(10,14)(11,17)(12,20)(22,24)(25,26)(27,30)(28,31)(29,32)(33,36)(34,37)(35,38)(39,40)(43,49)(47,51)(50,55)(54,56);(10,14)(12,20)(15,18)(19,23)(26,27)(31,33)(32,34)(38,39)(41,42,46,44)(43,47,49,51)(45,52,53,48)(50,56,55,54);(9,10,13,14)(11,18,17,15)(12,16,20,21)(19,24,23,22)(25,27,30,26)(28,31,36,33)(29,32,37,34)(35,39,40,38)(41,43)(42,47)(44,51)(45,50)(46,49)(48,54)(52,56)(53,55);(1,2)(3,5)(4,6)(7,8)(9,11)(10,15)(12,19)(13,17)(14,18)(16,22)(20,23)(21,24)(25,28,30,36)(26,31,27,33)(29,35,37,40)(32,38,34,39)(41,44,46,42)(43,47,49,51)(45,52,53,48)(50,54,55,56);(1,3,4,7)(2,5,6,8)(9,12)(10,16)(11,19)(13,20)(14,21)(15,22)(17,23)(18,24)(25,29)(26,32)(27,34)(28,35)(30,37)(31,38)(33,39)(36,40)(41,45,46,53)(42,48,44,52)(43,50,49,55)(47,54,51,56)Polycyclic presentation:
2^9|a^2=fhi,b^2=gh,c^2=h,d^2=fgh,e^2=fgi,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=efg,d^c=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5496380);