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 | |
1636 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)(7,13)(8,16)(9,10)(11,15)(12,17)(14,18)(19,20)(21,22,25,26)(23,29,31,27)(24,28,32,33)(30,35,36,34)(37,38,42,40)(39,45,47,43)(41,48,50,44)(46,51,49,52);(6,10)(8,17)(11,13)(14,20)(22,26)(24,32)(27,29)(30,36)(37,39,42,47)(38,43,40,45)(41,49,50,46)(44,52,48,51);(5,6)(7,13)(8,12)(9,10)(11,15)(14,19)(16,17)(18,20)(21,23,25,31)(22,27,26,29)(24,30,32,36)(28,34,33,35)(37,39)(38,43)(40,45)(41,46)(42,47)(44,51)(48,52)(49,50);(1,2)(3,4)(5,7,9,15)(6,11,10,13)(8,14,17,20)(12,18,16,19)(21,22)(23,29)(24,28)(25,26)(27,31)(30,35)(32,33)(34,36)(37,40,42,38)(39,43,47,45)(41,48,50,44)(46,51,49,52);(5,8,9,17)(6,12,10,16)(7,14,15,20)(11,18,13,19)(21,24)(22,28)(23,30)(25,32)(26,33)(27,34)(29,35)(31,36)(37,41)(38,44)(39,46)(40,48)(42,50)(43,51)(45,52)(47,49)Polycyclic presentation:
2^9|a^2=g,b^2=fgh,c^2=fh,d^2=h,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, 4377154);