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)(5,6,9,10)(7,13,15,11)(8,12,16,17)(14,19,20,18)(21,22)(23,28)(24,25)(26,27)(29,34)(30,36)(31,33)(32,35)(37,38,41,44)(39,45,47,42)(40,48,49,43)(46,50,52,51);(6,10)(8,16)(11,13)(14,20)(22,27)(25,29)(28,35)(31,36)(37,38,41,44)(39,45,47,42)(40,43,49,48)(46,51,52,50);(5,7,9,15)(6,11,10,13)(8,14,16,20)(12,18,17,19)(21,23)(22,28)(24,33)(25,31)(26,32)(27,35)(29,36)(30,34)(37,38)(39,42)(40,43)(41,44)(45,47)(46,50)(48,49)(51,52);(1,2)(3,4)(5,6)(7,13)(8,12)(9,10)(11,15)(14,19)(16,17)(18,20)(21,24)(22,29)(23,30)(25,27)(26,34)(28,31)(32,33)(35,36)(37,39,41,47)(38,42,44,45)(40,46,49,52)(43,50,48,51);(5,8,9,16)(6,12,10,17)(7,14,15,20)(11,18,13,19)(21,25,26,29)(22,24,27,34)(23,31,32,36)(28,33,35,30)(37,40)(38,43)(39,46)(41,49)(42,50)(44,48)(45,51)(47,52)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=gh,d^2=h,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, 4376860);