Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996375 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #45196 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1728 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(1,2)(3,4)(8,16)(11,17)(15,20)(18,19)(21,22,28,45,27,32,26,33)(23,34,41,29,37,51,38,48)(24,39,43,49,36,30,42,47)(25,31,40,52,44,46,35,50);(5,6,9,12)(7,13,14,10)(8,17,16,11)(15,18,20,19)(21,23)(22,29)(24,40)(25,43)(26,41)(27,37)(28,38)(30,31)(32,48)(33,51)(34,45)(35,36)(39,46)(42,44)(47,50)(49,52);(2,4)(5,7)(6,10)(8,15)(9,14)(11,18)(12,13)(16,20)(17,19)(21,24)(22,30)(23,35)(25,41)(26,42)(27,36)(28,43)(29,46)(31,48)(32,39)(33,49)(34,52)(37,40)(38,44)(45,47)(50,51);(7,14)(10,13)(15,20)(18,19)(21,25,26,35,27,44,28,40)(22,31,33,50,32,46,45,52)(23,36,38,43,37,24,41,42)(29,47,34,30,48,49,51,39);(5,8,9,16)(6,11,12,17)(7,15,14,20)(10,18,13,19)(22,32)(23,37)(24,36)(31,46)(33,45)(38,41)(42,43)(50,52)Polycyclic presentation:
2^9|a^2=fi,b^2=hi,d^2=f,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dhi,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7088132);