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 | |
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)(5,6,9,12)(7,13,14,10)(8,11,16,18)(15,19,20,17)(21,22)(23,25)(24,32)(26,27)(28,31)(29,33)(30,36)(34,35)(38,42)(39,45)(40,48)(50,52);(7,14)(10,13)(15,20)(17,19)(22,27)(24,33)(25,28)(30,34)(37,38,41,42)(39,46,45,43)(40,49,48,44)(47,50,51,52);(5,6)(7,10)(8,11)(9,12)(13,14)(15,17)(16,18)(19,20)(21,23)(22,28)(24,34)(25,27)(26,31)(29,35)(30,33)(32,36)(37,39,41,45)(38,43,42,46)(40,47,48,51)(44,50,49,52);(1,2,3,4)(5,7,9,14)(6,10,12,13)(8,15,16,20)(11,17,18,19)(21,24)(22,29)(23,30)(25,35)(26,33)(27,32)(28,36)(31,34)(37,38)(39,46)(40,44)(41,42)(43,45)(47,52)(48,49)(50,51);(5,8,9,16)(6,11,12,18)(7,15,14,20)(10,17,13,19)(21,25,26,28)(22,23,27,31)(24,35,33,36)(29,30,32,34)(37,40,41,48)(38,44,42,49)(39,47,45,51)(43,50,46,52)Polycyclic presentation:
2^9|a^2=fg,b^2=h,c^2=h,d^2=fi,e^2=g,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, 4407973);