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 | |
1556 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)(7,10,13,17)(8,15,16,11)(14,20,19,18)(21,22)(23,25)(24,31)(26,27)(28,32)(29,35)(30,36)(33,34)(37,38,42,45)(39,43,40,47)(41,44,49,52)(46,50,48,51);(7,13)(10,17)(14,19)(18,20)(22,27)(23,29)(24,32)(34,36)(37,39,42,40)(38,43,45,47)(41,48,49,46)(44,51,52,50);(5,7)(6,10)(8,14)(9,13)(11,18)(12,17)(15,20)(16,19)(21,23,26,29)(22,25,27,35)(24,33,32,30)(28,36,31,34)(37,40,42,39)(38,43,45,47)(41,48,49,46)(44,50,52,51);(1,2,3,4)(5,6)(7,10)(8,11)(9,12)(13,17)(14,18)(15,16)(19,20)(21,24)(22,28)(23,30)(25,34)(26,32)(27,31)(29,33)(35,36)(37,38)(39,43)(40,47)(41,44)(42,45)(46,50)(48,51)(49,52);(5,8,9,16)(6,11,12,15)(7,14,13,19)(10,18,17,20)(21,25)(22,23)(24,34)(26,35)(27,29)(28,30)(31,33)(32,36)(37,41)(38,44)(39,46)(40,48)(42,49)(43,50)(45,52)(47,51)Polycyclic presentation:
2^9|a^2=fgh,b^2=g,c^2=fg,d^2=ghi,e^2=fh,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, 5192016);