Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6469983 |
| 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×#535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2909 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)(5,6,9,12)(7,10,13,17)(8,15,16,11)(14,20,19,18)(21,22,26,29)(23,31,33,46)(24,28,36,44)(25,38,40,30)(27,39,42,34)(32,47,35,49)(37,52,51,45)(41,50,43,48);(5,7)(6,10)(8,14)(9,13)(11,18)(12,17)(15,20)(16,19)(21,23)(22,27)(24,35)(25,39)(26,33)(28,43)(29,42)(30,46)(31,38)(32,36)(34,40)(37,48)(41,44)(45,47)(49,52)(50,51);(2,4)(7,13)(10,17)(14,19)(18,20)(21,24,26,36)(22,28,29,44)(23,32,33,35)(25,37,40,51)(27,41,42,43)(30,45,38,52)(31,47,46,49)(34,48,39,50);(7,13)(10,17)(14,19)(18,20)(23,33)(24,36)(27,42)(28,44)(31,46)(34,39)(37,51)(45,52);(5,8,9,16)(6,11,12,15)(7,14,13,19)(10,18,17,20)(22,29)(24,36)(27,42)(30,38)(31,46)(32,35)(37,51)(48,50)Polycyclic presentation:
2^9|a^2=h,c^2=i,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,f^b=fi,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, 6403258);