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)(6,10)(7,13)(12,17)(15,19)(21,22)(23,27)(24,26)(25,28)(29,35)(30,31)(32,34)(33,36)(37,38,43,40)(39,44,41,47)(42,48,50,45)(46,52,49,51);(8,16)(12,17)(15,19)(18,20)(22,24)(27,31)(28,32)(35,36)(37,39,43,41)(38,44,40,47)(42,46,50,49)(45,51,48,52);(5,6)(7,14)(8,12)(9,10)(11,13)(15,20)(16,17)(18,19)(21,23)(22,27)(24,31)(25,33)(26,30)(28,36)(29,34)(32,35)(37,40,43,38)(39,44,41,47)(42,48,50,45)(46,51,49,52);(1,2,3,4)(5,7)(6,11)(8,15)(9,13)(10,14)(12,18)(16,19)(17,20)(21,24,26,22)(23,27,30,31)(25,32,34,28)(29,35,33,36)(37,41,43,39)(38,44,40,47)(42,49,50,46)(45,51,48,52);(5,8)(6,12)(7,15)(9,16)(10,17)(11,18)(13,19)(14,20)(21,25,26,34)(22,28,24,32)(23,29,30,33)(27,35,31,36)(37,42)(38,45)(39,46)(40,48)(41,49)(43,50)(44,51)(47,52)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=h,d^2=gi,e^2=f,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, 4989738);