Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4465744 |
| 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, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1652 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(2,4)(7,13)(8,15)(11,16)(12,17)(21,22,25,28)(23,29,30,26)(24,27,32,34)(31,35,36,33)(37,38)(39,45)(40,42)(41,44)(43,47)(46,51)(48,49)(50,52);(6,10)(11,16)(12,17)(18,20)(23,30)(26,29)(31,36)(33,35)(38,40)(41,49)(43,45)(46,52);(5,6,9,10)(7,11,13,16)(8,12,15,17)(14,18,19,20)(21,22)(23,26)(24,27)(25,28)(29,30)(31,33)(32,34)(35,36)(37,39,42,47)(38,43,40,45)(41,46,49,52)(44,50,48,51);(1,2)(3,4)(5,7,9,13)(6,11,10,16)(8,14,15,19)(12,18,17,20)(21,23)(22,26)(24,31)(25,30)(27,33)(28,29)(32,36)(34,35)(37,40,42,38)(39,43,47,45)(41,48,49,44)(46,50,52,51);(5,8)(6,12)(7,14)(9,15)(10,17)(11,18)(13,19)(16,20)(21,24,25,32)(22,27,28,34)(23,31,30,36)(26,33,29,35)(37,41)(38,44)(39,46)(40,48)(42,49)(43,50)(45,51)(47,52)Polycyclic presentation:
2^9|a^2=fg,c^2=fh,d^2=fh,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^b=ef,d^c=df68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4376569);