Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5605735 |
| 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:
(1,2,5,3)(4,7,8,6)(11,17)(12,19)(15,20)(16,21)(25,26)(27,30)(28,31)(29,32)(33,36)(34,37)(35,39)(38,40)(41,42,46,44)(43,49,50,47)(45,52,53,48)(51,54,56,55);(1,2,5,3)(4,6,8,7)(10,14)(15,20)(16,21)(22,24)(26,27)(29,37)(31,33)(35,38)(43,50)(45,53)(47,49)(48,52);(1,3,5,2)(4,7,8,6)(9,10)(11,15)(12,16)(13,14)(17,20)(18,22)(19,21)(23,24)(25,27,30,26)(28,33,36,31)(29,34,37,32)(35,40,38,39)(41,43,46,50)(42,47,44,49)(45,51,53,56)(48,54,52,55);(1,4,5,8)(2,6,3,7)(9,11)(10,15)(12,18)(13,17)(14,20)(16,22)(19,23)(21,24)(25,28,30,36)(26,31,27,33)(29,38,37,35)(32,40,34,39)(41,44,46,42)(43,47,50,49)(45,52,53,48)(51,54,56,55);(9,12,13,19)(10,16,14,21)(11,18,17,23)(15,22,20,24)(25,29,30,37)(26,32,27,34)(28,35,36,38)(31,39,33,40)(41,45)(42,48)(43,51)(44,52)(46,53)(47,54)(49,55)(50,56)Polycyclic presentation:
2^9|a^2=fhi,b^2=ghi,c^2=hi,d^2=hi,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5862068);