Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6433044 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2016 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)(5,6)(7,10)(8,12)(9,11)(13,16)(14,17)(15,18)(19,20)(21,22,27,32)(23,33,37,43)(24,29,40,47)(25,31,26,30)(28,35,44,36)(34,49,38,52)(39,48,41,46)(42,51,45,50);(5,7)(6,10)(8,14)(9,13)(11,16)(12,17)(15,19)(18,20)(21,23,25,35,27,37,26,36)(22,28,30,43,32,44,31,33)(24,38,41,50,40,34,39,51)(29,45,48,49,47,42,46,52);(1,2,3,4)(7,13)(10,16)(14,19)(17,20)(21,24)(22,29)(23,34)(25,41)(26,39)(27,40)(28,42)(30,48)(31,46)(32,47)(33,49)(35,51)(36,50)(37,38)(43,52)(44,45);(6,11)(10,16)(12,18)(17,20)(21,25,27,26)(22,30,32,31)(23,35,37,36)(24,39,40,41)(28,43,44,33)(29,46,47,48)(34,50,38,51)(42,49,45,52);(5,8,9,15)(6,12,11,18)(7,14,13,19)(10,17,16,20)(24,40)(29,47)(34,38)(39,41)(42,45)(46,48)(49,52)(50,51)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=g,d^2=i,e^2=hi,f^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,c^b=ch,d^c=di,e^c=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6344443);