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)(6,10)(7,13)(8,15)(18,20)(21,22,26,23)(24,29,32,27)(25,30,33,28)(31,34,36,35)(37,38,41,44)(39,42,45,49)(40,47,48,43)(46,52,51,50);(6,10)(7,13)(12,17)(14,19)(22,23)(25,33)(27,29)(31,36)(39,45)(42,49)(46,51)(50,52);(5,6)(7,11)(8,12)(9,10)(13,16)(14,18)(15,17)(19,20)(21,23,26,22)(24,27,32,29)(25,30,33,28)(31,34,36,35)(37,39,41,45)(38,42,44,49)(40,46,48,51)(43,50,47,52);(1,2)(3,4)(5,7,9,13)(6,11,10,16)(8,14,15,19)(12,18,17,20)(21,24,26,32)(22,27,23,29)(25,31,33,36)(28,34,30,35)(37,38)(39,42)(40,43)(41,44)(45,49)(46,50)(47,48)(51,52);(5,8)(6,12)(7,14)(9,15)(10,17)(11,18)(13,19)(16,20)(21,25)(22,28)(23,30)(24,31)(26,33)(27,34)(29,35)(32,36)(37,40,41,48)(38,43,44,47)(39,46,45,51)(42,50,49,52)Polycyclic presentation:
2^9|a^2=gh,c^2=gh,d^2=fg,e^2=fgh,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, 4381316);