Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #6132643 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C4 |
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:
(1,2,5,6)(3,4,7,8)(10,14)(11,17)(16,21)(19,23)(25,26)(27,33)(28,32)(29,30)(31,35)(34,39)(36,37)(38,40)(44,51)(45,53)(47,49)(48,50);(3,7)(4,8)(10,14)(11,17)(12,20)(22,24)(26,30)(31,33)(32,36)(38,39)(41,42,46,43)(44,49,51,47)(45,48,53,50)(52,55,56,54);(1,3)(2,4)(5,7)(6,8)(9,10,13,14)(11,18,17,15)(12,16,20,21)(19,24,23,22)(25,26)(27,33)(28,36)(29,30)(31,35)(32,37)(34,38)(39,40)(41,43,46,42)(44,49,51,47)(45,50,53,48)(52,55,56,54);(1,4)(2,3)(5,8)(6,7)(9,11,13,17)(10,15,14,18)(12,19,20,23)(16,22,21,24)(25,27,29,35)(26,31,30,33)(28,34,37,40)(32,38,36,39)(41,44)(42,47)(43,49)(45,52)(46,51)(48,54)(50,55)(53,56);(9,12)(10,16)(11,19)(13,20)(14,21)(15,22)(17,23)(18,24)(25,28,29,37)(26,32,30,36)(27,34,35,40)(31,38,33,39)(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=hi,c^2=ghi,d^2=fg,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dfi,e^b=eg,d^c=dfg,e^c=ef62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6178247);