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 | |
1570 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)(5,6,9,12)(7,13,15,10)(8,11,16,18)(14,19,20,17)(21,22,26,27)(23,30,33,28)(24,25,34,29)(31,35,36,32)(37,38,41,45)(39,43,47,42)(40,44,49,48)(46,51,52,50);(8,16)(11,18)(14,20)(17,19)(22,27)(25,29)(28,30)(32,35)(37,39,41,47)(38,42,45,43)(40,46,49,52)(44,50,48,51);(5,7,9,15)(6,10,12,13)(8,14,16,20)(11,17,18,19)(21,23,26,33)(22,28,27,30)(24,31,34,36)(25,32,29,35)(37,39)(38,43)(40,46)(41,47)(42,45)(44,51)(48,50)(49,52);(1,2)(3,4)(5,6)(7,13)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20)(21,24,26,34)(22,29,27,25)(23,31,33,36)(28,35,30,32)(37,38)(39,42)(40,48)(41,45)(43,47)(44,49)(46,51)(50,52);(5,8)(6,11)(7,14)(9,16)(10,17)(12,18)(13,19)(15,20)(21,25,26,29)(22,24,27,34)(23,32,33,35)(28,31,30,36)(37,40,41,49)(38,44,45,48)(39,46,47,52)(42,50,43,51)Polycyclic presentation:
2^9|a^2=gh,b^2=h,c^2=g,d^2=fh,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, 5524379);