Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| 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 | |
1538 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)(5,6,10,8)(7,11,13,14)(9,16,17,12)(15,20,19,18)(21,22,26,27)(23,24,29,32)(25,28,34,35)(30,31,33,36)(37,38,43,41)(39,44,40,47)(42,45,50,49)(46,51,48,52);(7,13)(11,14)(15,19)(18,20)(22,27)(23,29)(28,35)(30,33)(37,39,43,40)(38,44,41,47)(42,46,50,48)(45,51,49,52);(5,7)(6,11)(8,14)(9,15)(10,13)(12,18)(16,20)(17,19)(21,23)(22,24)(25,33)(26,29)(27,32)(28,36)(30,34)(31,35)(37,40,43,39)(38,44,41,47)(42,48,50,46)(45,51,49,52);(1,2)(3,4)(5,8,10,6)(7,14,13,11)(9,16,17,12)(15,20,19,18)(21,24)(22,23)(25,31)(26,32)(27,29)(28,30)(33,35)(34,36)(37,41,43,38)(39,44,40,47)(42,49,50,45)(46,51,48,52);(5,9,10,17)(6,12,8,16)(7,15,13,19)(11,18,14,20)(21,25,26,34)(22,28,27,35)(23,30,29,33)(24,31,32,36)(37,42,43,50)(38,45,41,49)(39,46,40,48)(44,51,47,52)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=g,d^2=fgh,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4513476);