Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | C2×#3455 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3018 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)(7,13)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20)(21,22,25,29,26,30,36,42)(23,27,33,39,31,37,44,48)(24,28,35,41,34,40,46,50)(32,38,45,49,43,47,51,52);(5,7)(6,10)(8,14)(9,15)(11,17)(12,13)(16,20)(18,19)(21,23)(22,27)(24,32)(25,33)(26,31)(28,38)(29,39)(30,37)(34,43)(35,45)(36,44)(40,47)(41,49)(42,48)(46,51)(50,52);(2,4)(5,8,9,16)(6,11,12,18)(7,14,15,20)(10,17,13,19)(23,31)(27,37)(32,43)(33,44)(38,47)(39,48)(45,51)(49,52);(7,15)(10,13)(14,20)(17,19)(21,24)(22,28)(23,32)(25,35)(26,34)(27,38)(29,41)(30,40)(31,43)(33,45)(36,46)(37,47)(39,49)(42,50)(44,51)(48,52);(6,12)(10,13)(11,18)(17,19)(23,31)(24,34)(27,37)(28,40)(33,44)(35,46)(39,48)(41,50)Polycyclic presentation:
2^9|a^2=h,c^2=f,h^2=i,b^a=bf,c^a=cg,e^a=ef,c^b=ci,d^b=df,e^b=ei,e^d=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6293297);