Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6469956 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3455 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2930 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 52 points:
(1,2)(3,4)(5,6)(7,10)(8,11)(9,12)(13,17)(14,18)(15,19)(16,20)(21,22,27,30)(23,32,35,43)(24,29,36,45)(25,31,26,39)(28,40,42,34)(33,46,47,51)(37,44,38,49)(41,50,52,48);(5,7)(6,10)(8,15)(9,13)(11,19)(12,17)(14,16)(18,20)(21,23)(22,28)(24,33)(25,34)(26,40)(27,35)(29,41)(30,42)(31,43)(32,39)(36,47)(37,48)(38,50)(44,51)(45,52)(46,49);(2,4)(7,13)(10,17)(14,15)(18,19)(21,24)(22,29)(23,33)(25,37)(26,38)(27,36)(28,41)(30,45)(31,44)(32,46)(34,48)(35,47)(39,49)(40,50)(42,52)(43,51);(5,8,9,16)(6,11,12,20)(7,14,13,15)(10,18,17,19)(22,30)(24,36)(28,42)(31,39)(32,43)(33,47)(37,38)(48,50);(6,12)(10,17)(11,20)(18,19)(21,25,27,26)(22,31,30,39)(23,34,35,40)(24,37,36,38)(28,43,42,32)(29,44,45,49)(33,48,47,50)(41,51,52,46)Polycyclic presentation:
2^9|a^2=i,d^2=h,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fi,c^b=ch,d^b=dh,f^b=fi,d^c=di74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6388727);