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 | |
1627 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)(5,6,9,10)(7,13,14,11)(8,12,16,17)(15,19,20,18)(21,22)(23,27)(24,26)(25,32)(28,34)(29,36)(30,31)(33,35)(37,38)(39,45)(40,43)(41,44)(42,47)(46,51)(48,49)(50,52);(6,10)(7,14)(8,16)(18,19)(22,24)(27,31)(28,32)(35,36)(37,38,41,44)(39,45,47,42)(40,43,49,48)(46,51,52,50);(5,6)(7,11)(8,17)(9,10)(12,16)(13,14)(15,19)(18,20)(21,23,26,30)(22,27,24,31)(25,33,34,29)(28,36,32,35)(37,38)(39,45)(40,48)(41,44)(42,47)(43,49)(46,50)(51,52);(1,2,3,4)(5,7,9,14)(6,11,10,13)(8,15,16,20)(12,18,17,19)(21,24,26,22)(23,27,30,31)(25,32,34,28)(29,35,33,36)(37,39,41,47)(38,42,44,45)(40,46,49,52)(43,50,48,51);(5,8)(6,12)(7,15)(9,16)(10,17)(11,18)(13,19)(14,20)(21,25,26,34)(22,28,24,32)(23,29,30,33)(27,35,31,36)(37,40,41,49)(38,43,44,48)(39,46,47,52)(42,50,45,51)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=fh,d^2=i,e^2=fg,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, 5188047);