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, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1654 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)(21,22,25,26)(23,29,30,27)(24,28,32,33)(31,35,36,34)(37,38)(39,43)(40,42)(41,48)(44,50)(45,52)(46,47)(49,51);(6,10)(11,16)(12,17)(18,20)(22,26)(23,30)(24,32)(34,35)(38,40)(43,47)(44,48)(51,52);(5,6)(7,11)(8,12)(9,10)(13,16)(14,18)(15,17)(19,20)(21,22)(23,27)(24,33)(25,26)(28,32)(29,30)(31,35)(34,36)(37,39,42,46)(38,43,40,47)(41,49,50,45)(44,52,48,51);(1,2,3,4)(5,7)(6,11)(8,14)(9,13)(10,16)(12,18)(15,19)(17,20)(21,23,25,30)(22,27,26,29)(24,31,32,36)(28,34,33,35)(37,40,42,38)(39,43,46,47)(41,48,50,44)(45,51,49,52);(5,8,9,15)(6,12,10,17)(7,14,13,19)(11,18,16,20)(21,24)(22,28)(23,31)(25,32)(26,33)(27,34)(29,35)(30,36)(37,41,42,50)(38,44,40,48)(39,45,46,49)(43,51,47,52)Polycyclic presentation:
2^9|a^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, 5190927);