Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| 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:
(4,8)(6,7)(10,14)(11,17)(16,21)(18,23)(25,26,29,32)(27,30,33,37)(28,35,36,31)(34,40,39,38)(41,42)(43,48)(44,49)(45,50)(46,47)(51,56)(52,53)(54,55);(1,2,5,3)(4,6,8,7)(11,17)(15,20)(18,23)(22,24)(27,33)(30,37)(34,39)(38,40)(42,47)(43,49)(50,55)(51,53);(1,3,5,2)(4,7,8,6)(9,10,13,14)(11,15,17,20)(12,16,19,21)(18,22,23,24)(25,27,29,33)(26,30,32,37)(28,34,36,39)(31,38,35,40)(41,43,46,49)(42,48,47,44)(45,53,54,51)(50,52,55,56);(1,4,5,8)(2,6,3,7)(9,11,13,17)(10,15,14,20)(12,18,19,23)(16,22,21,24)(25,26)(27,30)(28,31)(29,32)(33,37)(34,38)(35,36)(39,40)(41,44,46,48)(42,49,47,43)(45,52,54,56)(50,53,55,51);(9,12,13,19)(10,16,14,21)(11,18,17,23)(15,22,20,24)(25,28,29,36)(26,31,32,35)(27,34,33,39)(30,38,37,40)(41,45)(42,50)(43,51)(44,52)(46,54)(47,55)(48,56)(49,53)Polycyclic presentation:
2^9|a^2=fh,b^2=fghi,c^2=fi,d^2=hi,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4636979);