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) = C27 |
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 | |
1538 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 52 points:
(2,4)(7,13)(8,15)(11,16)(12,17)(21,22,26,27)(23,28,29,24)(25,33,34,30)(31,32,36,35)(37,38,43,46)(39,44,40,41)(42,45,50,52)(47,51,48,49);(6,10)(11,16)(12,17)(18,20)(22,27)(23,29)(25,34)(32,35)(37,39,43,40)(38,41,46,44)(42,48,50,47)(45,51,52,49);(5,6)(7,11)(8,12)(9,10)(13,16)(14,18)(15,17)(19,20)(21,23)(22,28)(24,27)(25,31)(26,29)(30,35)(32,33)(34,36)(37,40,43,39)(38,44,46,41)(42,48,50,47)(45,51,52,49);(1,2,3,4)(5,7)(6,11)(8,14)(9,13)(10,16)(12,18)(15,19)(17,20)(21,24,26,28)(22,29,27,23)(25,32,34,35)(30,36,33,31)(37,41,43,44)(38,40,46,39)(42,49,50,51)(45,48,52,47);(5,8,9,15)(6,12,10,17)(7,14,13,19)(11,18,16,20)(21,25,26,34)(22,30,27,33)(23,31,29,36)(24,32,28,35)(37,42,43,50)(38,45,46,52)(39,47,40,48)(41,49,44,51)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=g,d^2=fghi,e^2=gh,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, 4410213);