Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| 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 | |
1556 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)(12,17)(15,19)(21,22,25,26)(23,27,31,29)(24,32,34,28)(30,36,33,35)(39,45)(41,49)(43,46)(44,48);(8,16)(12,17)(15,19)(18,20)(22,26)(27,29)(28,32)(35,36)(37,38,42,40)(39,46,45,43)(41,48,49,44)(47,50,51,52);(5,6)(7,14)(8,12)(9,10)(11,13)(15,20)(16,17)(18,19)(21,23)(22,27)(24,33)(25,31)(26,29)(28,36)(30,34)(32,35)(37,39,42,45)(38,43,40,46)(41,47,49,51)(44,50,48,52);(1,2,3,4)(5,7)(6,11)(8,15)(9,13)(10,14)(12,18)(16,19)(17,20)(21,22)(23,29)(24,28)(25,26)(27,31)(30,36)(32,34)(33,35)(37,40,42,38)(39,43,45,46)(41,48,49,44)(47,50,51,52);(5,8,9,16)(6,12,10,17)(7,15,13,19)(11,18,14,20)(21,24)(22,28)(23,30)(25,34)(26,32)(27,35)(29,36)(31,33)(37,41,42,49)(38,44,40,48)(39,47,45,51)(43,50,46,52)Polycyclic presentation:
2^9|a^2=f,b^2=h,c^2=h,d^2=fgi,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5188196);