Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| 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)(7,10)(8,13)(9,12)(11,15)(14,17)(16,19)(18,20)(21,22,25,26)(23,27,31,29)(24,32,34,28)(30,36,33,35)(38,42)(39,45)(40,47)(51,52);(6,7)(8,15)(9,17)(18,19)(22,26)(27,29)(28,32)(35,36)(37,38,41,42)(39,43,45,50)(40,48,47,44)(46,52,49,51);(5,7,10,6)(8,13,15,11)(9,12,17,14)(16,18,20,19)(21,23)(22,27)(24,33)(25,31)(26,29)(28,36)(30,34)(32,35)(37,39)(38,43)(40,49)(41,45)(42,50)(44,52)(46,47)(48,51);(1,2,3,4)(5,8)(6,11)(7,13)(9,16)(10,15)(12,18)(14,19)(17,20)(21,22)(23,29)(24,28)(25,26)(27,31)(30,36)(32,34)(33,35)(37,38)(39,43)(40,44)(41,42)(45,50)(46,51)(47,48)(49,52);(5,9,10,17)(6,12,7,14)(8,16,15,20)(11,18,13,19)(21,24)(22,28)(23,30)(25,34)(26,32)(27,35)(29,36)(31,33)(37,40,41,47)(38,44,42,48)(39,46,45,49)(43,51,50,52)Polycyclic presentation:
2^9|a^2=fg,b^2=gh,c^2=h,d^2=fi,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5188036);