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 | |
1652 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)(23,29)(24,32)(25,26)(27,31)(28,33)(30,34)(35,36)(37,38,41,44)(39,42,45,49)(40,47,48,43)(46,52,51,50);(6,10)(7,13)(12,17)(14,19)(22,26)(24,33)(27,29)(30,36)(39,45)(42,49)(46,51)(50,52);(5,6,9,10)(7,11,13,16)(8,12,15,17)(14,18,19,20)(21,22)(23,29)(24,28)(25,26)(27,31)(30,35)(32,33)(34,36)(37,39)(38,42)(40,46)(41,45)(43,50)(44,49)(47,52)(48,51);(1,2)(3,4)(5,7)(6,11)(8,14)(9,13)(10,16)(12,18)(15,19)(17,20)(21,23,25,31)(22,27,26,29)(24,30,33,36)(28,34,32,35)(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)(21,24,25,33)(22,28,26,32)(23,30,31,36)(27,34,29,35)(37,40,41,48)(38,43,44,47)(39,46,45,51)(42,50,49,52)Polycyclic presentation:
2^9|a^2=fgh,c^2=g,d^2=f,e^2=gh,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, 4378997);