Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| 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 | |
1566 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 52 points:
(1,2,3,4)(8,15)(12,17)(13,19)(18,20)(22,26)(24,31)(27,32)(30,35)(38,39)(40,47)(41,49)(50,51);(6,10)(11,16)(12,17)(18,20)(22,26)(23,29)(28,33)(30,35)(37,38,42,39)(40,45,47,43)(41,44,49,46)(48,51,52,50);(5,6)(7,11)(8,12)(9,10)(13,18)(14,16)(15,17)(19,20)(21,22)(23,27)(24,28)(25,26)(29,32)(30,34)(31,33)(35,36)(37,39,42,38)(40,45,47,43)(41,44,49,46)(48,50,52,51);(1,2)(3,4)(5,7,9,14)(6,11,10,16)(8,13,15,19)(12,18,17,20)(21,23)(22,27)(24,30)(25,29)(26,32)(28,34)(31,35)(33,36)(37,40)(38,43)(39,45)(41,48)(42,47)(44,50)(46,51)(49,52);(5,8)(6,12)(7,13)(9,15)(10,17)(11,18)(14,19)(16,20)(21,24,25,31)(22,28,26,33)(23,30,29,35)(27,34,32,36)(37,41,42,49)(38,44,39,46)(40,48,47,52)(43,50,45,51)Polycyclic presentation:
2^9|a^2=fi,b^2=f,c^2=f,d^2=fgh,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4516139);