Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| 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 | |
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)(5,6,10,8)(7,11,13,14)(9,16,17,12)(15,20,19,18)(21,22)(23,24)(25,29)(26,27)(28,32)(30,31)(33,36)(34,35)(37,38,42,45)(39,43,40,46)(41,44,49,51)(47,50,48,52);(7,13)(11,14)(15,19)(18,20)(22,27)(23,28)(29,35)(30,33)(37,39,42,40)(38,43,45,46)(41,47,49,48)(44,50,51,52);(5,7)(6,11)(8,14)(9,15)(10,13)(12,18)(16,20)(17,19)(21,23,26,28)(22,24,27,32)(25,33,34,30)(29,36,35,31)(37,40,42,39)(38,43,45,46)(41,48,49,47)(44,50,51,52);(1,2)(3,4)(5,8,10,6)(7,14,13,11)(9,16,17,12)(15,20,19,18)(21,24)(22,28)(23,27)(25,31)(26,32)(29,33)(30,35)(34,36)(37,38)(39,46)(40,43)(41,44)(42,45)(47,52)(48,50)(49,51);(5,9,10,17)(6,12,8,16)(7,15,13,19)(11,18,14,20)(21,25,26,34)(22,29,27,35)(23,30,28,33)(24,31,32,36)(37,41,42,49)(38,44,45,51)(39,47,40,48)(43,50,46,52)Polycyclic presentation:
2^9|a^2=fgh,b^2=g,c^2=fg,d^2=fh,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4513225);