Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5035983 |
| 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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1556 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 52 points:
(2,4)(5,6,9,10)(7,11,15,13)(8,12,17,18)(14,19,16,20)(21,22)(23,29)(24,31)(25,28)(26,27)(30,36)(32,34)(33,35)(37,38,41,44)(39,45,47,42)(40,48,50,43)(46,51,49,52);(6,10)(11,13)(12,18)(19,20)(22,27)(24,29)(25,33)(30,34)(37,39,41,47)(38,42,44,45)(40,49,50,46)(43,52,48,51);(5,7,9,15)(6,11,10,13)(8,16,17,14)(12,20,18,19)(21,23)(22,24)(25,34)(26,31)(27,29)(28,36)(30,33)(32,35)(37,39)(38,42)(40,46)(41,47)(43,51)(44,45)(48,52)(49,50);(1,2,3,4)(5,6)(7,13)(8,12)(9,10)(11,15)(14,20)(16,19)(17,18)(21,24)(22,23)(25,32)(26,29)(27,31)(28,30)(33,36)(34,35)(37,40)(38,43)(39,46)(41,50)(42,51)(44,48)(45,52)(47,49);(5,8,9,17)(6,12,10,18)(7,14,15,16)(11,19,13,20)(21,25,26,33)(22,28,27,35)(23,30,31,34)(24,32,29,36)(37,38)(39,45)(40,43)(41,44)(42,47)(46,52)(48,50)(49,51)Polycyclic presentation:
2^9|a^2=fgh,b^2=h,c^2=fg,d^2=fi,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4990976);