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)(7,13)(8,15)(11,16)(12,17)(21,22,25,28)(23,29,30,26)(24,27,32,34)(31,35,36,33)(37,38,42,46)(39,44,40,43)(41,45,49,52)(47,51,48,50);(6,10)(11,16)(12,17)(18,20)(23,30)(26,29)(31,36)(33,35)(37,39,42,40)(38,43,46,44)(41,48,49,47)(45,51,52,50);(5,6)(7,11)(8,12)(9,10)(13,16)(14,18)(15,17)(19,20)(21,22)(23,26)(24,27)(25,28)(29,30)(31,33)(32,34)(35,36)(37,40,42,39)(38,44,46,43)(41,48,49,47)(45,51,52,50);(1,2)(3,4)(5,7,9,13)(6,11,10,16)(8,14,15,19)(12,18,17,20)(21,23)(22,26)(24,31)(25,30)(27,33)(28,29)(32,36)(34,35)(37,38)(39,44)(40,43)(41,45)(42,46)(47,51)(48,50)(49,52);(5,8)(6,12)(7,14)(9,15)(10,17)(11,18)(13,19)(16,20)(21,24,25,32)(22,27,28,34)(23,31,30,36)(26,33,29,35)(37,41)(38,45)(39,47)(40,48)(42,49)(43,50)(44,51)(46,52)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=f,d^2=h,e^2=fg,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, 4375330);