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)(21,22,26,27)(23,24,28,32)(25,33,34,29)(30,36,35,31)(37,38,42,46)(39,44,40,43)(41,45,49,52)(47,51,48,50);(6,10)(11,16)(12,17)(18,20)(22,27)(23,28)(25,34)(31,36)(37,39,42,40)(38,43,46,44)(41,48,49,47)(45,51,52,50);(5,6,9,10)(7,11,13,16)(8,12,15,17)(14,18,19,20)(21,23)(22,24)(25,30)(26,28)(27,32)(29,31)(33,36)(34,35)(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)(21,24)(22,28)(23,27)(25,31)(26,32)(29,35)(30,33)(34,36)(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)(21,25,26,34)(22,29,27,33)(23,30,28,35)(24,31,32,36)(37,41,42,49)(38,45,46,52)(39,47,40,48)(43,50,44,51)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=fh,d^2=fgh,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, 4376433);