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 | |
1609 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)(23,29)(24,26)(25,28)(27,31)(30,35)(32,33)(34,36)(37,38)(39,43)(40,42)(41,48)(44,47)(45,49)(46,51)(50,52);(6,10)(11,16)(12,17)(18,20)(22,24)(25,33)(27,29)(30,36)(37,39,42,47)(38,43,40,44)(41,46,49,52)(45,50,48,51);(5,6,9,10)(7,11,13,16)(8,12,15,17)(14,18,19,20)(21,23,26,31)(22,27,24,29)(25,30,33,36)(28,34,32,35)(37,39)(38,44)(40,43)(41,46)(42,47)(45,51)(48,50)(49,52);(1,2)(3,4)(5,7,9,13)(6,11,10,16)(8,14,15,19)(12,18,17,20)(21,24,26,22)(23,27,31,29)(25,32,33,28)(30,34,36,35)(37,40,42,38)(39,43,47,44)(41,48,49,45)(46,50,52,51);(5,8)(6,12)(7,14)(9,15)(10,17)(11,18)(13,19)(16,20)(21,25)(22,28)(23,30)(24,32)(26,33)(27,34)(29,35)(31,36)(37,41,42,49)(38,45,40,48)(39,46,47,52)(43,50,44,51)Polycyclic presentation:
2^9|b^2=fg,c^2=gh,d^2=fh,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, 4376726);