Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4465744 |
| automorphism | Aut(G) order 2097152 |
| 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 | |
1554 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)(6,10)(7,13)(8,15)(18,20)(21,22,26,24)(23,27,29,30)(25,32,33,28)(31,36,35,34)(37,38,42,40)(39,44,47,43)(41,48,50,45)(46,51,49,52);(6,10)(7,13)(12,17)(14,19)(23,29)(27,30)(31,35)(34,36)(37,39,42,47)(38,43,40,44)(41,49,50,46)(45,52,48,51);(5,6,9,10)(7,11,13,16)(8,12,15,17)(14,18,19,20)(21,23)(22,27)(24,30)(25,31)(26,29)(28,34)(32,36)(33,35)(37,39)(38,44)(40,43)(41,46)(42,47)(45,52)(48,51)(49,50);(1,2)(3,4)(5,7)(6,11)(8,14)(9,13)(10,16)(12,18)(15,19)(17,20)(21,24,26,22)(23,30,29,27)(25,32,33,28)(31,36,35,34)(37,40,42,38)(39,44,47,43)(41,48,50,45)(46,52,49,51);(5,8)(6,12)(7,14)(9,15)(10,17)(11,18)(13,19)(16,20)(21,25)(22,28)(23,31)(24,32)(26,33)(27,34)(29,35)(30,36)(37,41,42,50)(38,45,40,48)(39,46,47,49)(43,51,44,52)Polycyclic presentation:
2^9|a^2=gh,b^2=f,c^2=g,d^2=gh,e^2=f,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, 4375540);