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 | |
1625 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)(6,10)(7,13)(8,15)(18,20)(21,22,26,27)(23,25,29,35)(24,31,34,28)(30,36,32,33)(37,38)(39,44)(40,47)(41,43)(42,45)(46,51)(48,52)(49,50);(6,10)(7,13)(12,17)(14,19)(22,27)(25,35)(28,31)(33,36)(37,39,43,40)(38,44,41,47)(42,46,50,48)(45,51,49,52);(5,6)(7,11)(8,12)(9,10)(13,16)(14,18)(15,17)(19,20)(21,23,26,29)(22,25,27,35)(24,32,34,30)(28,36,31,33)(37,40,43,39)(38,44,41,47)(42,48,50,46)(45,51,49,52);(1,2,3,4)(5,7,9,13)(6,11,10,16)(8,14,15,19)(12,18,17,20)(21,24,26,34)(22,28,27,31)(23,30,29,32)(25,33,35,36)(37,41,43,38)(39,44,40,47)(42,49,50,45)(46,51,48,52);(5,8,9,15)(6,12,10,17)(7,14,13,19)(11,18,16,20)(21,25)(22,29)(23,27)(24,33)(26,35)(28,32)(30,31)(34,36)(37,42,43,50)(38,45,41,49)(39,46,40,48)(44,51,47,52)Polycyclic presentation:
2^9|a^2=f,b^2=gh,c^2=fgh,d^2=i,e^2=g,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, 4408033);