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)(5,6,9,10)(7,13,14,11)(8,16,17,12)(15,18,20,19)(21,22,26,29)(23,30,27,24)(25,33,34,28)(31,32,35,36)(38,40)(41,49)(43,45)(46,52);(6,10)(7,14)(8,17)(18,19)(23,27)(24,30)(31,35)(32,36)(37,38,42,40)(39,45,47,43)(41,48,49,44)(46,50,52,51);(5,7,9,14)(6,11,10,13)(8,15,17,20)(12,18,16,19)(21,23,26,27)(22,24,29,30)(25,31,34,35)(28,32,33,36)(37,39,42,47)(38,43,40,45)(41,46,49,52)(44,50,48,51);(1,2)(3,4)(5,6)(7,13)(8,12)(9,10)(11,14)(15,19)(16,17)(18,20)(21,24)(22,27)(23,29)(25,32)(26,30)(28,35)(31,33)(34,36)(37,40,42,38)(39,43,47,45)(41,48,49,44)(46,50,52,51);(5,8,9,17)(6,12,10,16)(7,15,14,20)(11,18,13,19)(21,25)(22,28)(23,31)(24,32)(26,34)(27,35)(29,33)(30,36)(37,41,42,49)(38,44,40,48)(39,46,47,52)(43,50,45,51)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=h,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, 4376956);