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 | |
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)(5,6,10,8)(7,13,15,11)(9,12,17,16)(14,19,20,18)(21,22,26,27)(23,25,31,28)(24,32,34,29)(30,33,36,35)(39,45)(40,47)(42,48)(43,50);(9,17)(12,16)(14,20)(18,19)(22,27)(25,28)(29,32)(33,35)(37,38,41,44)(39,42,45,48)(40,43,47,50)(46,49,51,52);(5,7,10,15)(6,11,8,13)(9,14,17,20)(12,18,16,19)(21,23)(22,28)(24,30)(25,27)(26,31)(29,33)(32,35)(34,36)(37,38)(39,42)(40,43)(41,44)(45,48)(46,49)(47,50)(51,52);(1,2)(3,4)(5,8,10,6)(7,11,15,13)(9,16,17,12)(14,18,20,19)(21,24,26,34)(22,29,27,32)(23,30,31,36)(25,35,28,33)(37,39,41,45)(38,42,44,48)(40,46,47,51)(43,49,50,52);(5,9,10,17)(6,12,8,16)(7,14,15,20)(11,18,13,19)(21,25,26,28)(22,23,27,31)(24,33,34,35)(29,36,32,30)(37,40)(38,43)(39,46)(41,47)(42,49)(44,50)(45,51)(48,52)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=fg,d^2=gh,e^2=g,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, 5524103);