Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5605735 |
| 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)(7,13)(8,12)(9,10)(11,15)(14,20)(16,19)(17,18)(21,22,25,28)(23,29,30,26)(24,32,33,27)(31,34,36,35)(37,38)(39,46)(40,44)(41,43)(42,49)(45,50)(47,51)(48,52);(6,10)(11,13)(12,18)(19,20)(23,30)(24,33)(26,29)(27,32)(37,39,43,40)(38,44,41,46)(42,47,50,48)(45,51,49,52);(5,6)(7,13)(8,12)(9,10)(11,15)(14,20)(16,19)(17,18)(21,23)(22,26)(24,31)(25,30)(27,34)(28,29)(32,35)(33,36)(37,40,43,39)(38,44,41,46)(42,48,50,47)(45,51,49,52);(1,2)(3,4)(5,7,9,15)(6,11,10,13)(8,16,17,14)(12,20,18,19)(21,22)(23,29)(24,27)(25,28)(26,30)(31,35)(32,33)(34,36)(37,41,43,38)(39,46,40,44)(42,45,50,49)(47,51,48,52);(5,8,9,17)(6,12,10,18)(7,14,15,16)(11,19,13,20)(21,24)(22,27)(23,31)(25,33)(26,34)(28,32)(29,35)(30,36)(37,42,43,50)(38,45,41,49)(39,47,40,48)(44,51,46,52)Polycyclic presentation:
2^9|a^2=g,b^2=fgh,c^2=fgh,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5519329);