Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5473759 |
| 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 | |
1540 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)(5,6,10,7)(8,13,16,11)(9,12,17,14)(15,19,20,18)(21,22,25,28)(23,26,29,33)(24,31,32,27)(30,36,35,34)(39,45)(43,47)(46,51)(50,52);(6,7)(9,17)(11,13)(15,20)(23,29)(26,33)(30,35)(34,36)(37,38,42,40)(39,43,45,47)(41,48,49,44)(46,52,51,50);(5,7,10,6)(8,13,16,11)(9,12,17,14)(15,18,20,19)(21,23,25,29)(22,26,28,33)(24,30,32,35)(27,34,31,36)(37,39)(38,43)(40,47)(41,46)(42,45)(44,50)(48,52)(49,51);(1,2)(3,4)(5,8,10,16)(6,11,7,13)(9,15,17,20)(12,18,14,19)(21,22)(23,26)(24,27)(25,28)(29,33)(30,34)(31,32)(35,36)(37,40,42,38)(39,43,45,47)(41,48,49,44)(46,50,51,52);(5,9)(6,12)(7,14)(8,15)(10,17)(11,18)(13,19)(16,20)(21,24,25,32)(22,27,28,31)(23,30,29,35)(26,34,33,36)(37,41,42,49)(38,44,40,48)(39,46,45,51)(43,50,47,52)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=h,d^2=fg,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=efg,d^c=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5389107);