Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993947 |
| 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 | C2×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1661 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)(8,16)(11,17)(15,20)(18,19)(21,22,27,32,28,33,26,42)(23,29,36,45,37,46,35,50)(24,38,39,47,25,31,41,30)(34,44,49,52,40,48,51,43);(1,2,3,4)(5,6,9,12)(7,13,14,10)(8,17,16,11)(15,18,20,19)(21,23)(22,29)(24,34)(25,40)(26,35)(27,36)(28,37)(30,43)(31,48)(32,45)(33,46)(38,44)(39,49)(41,51)(42,50)(47,52);(5,7)(6,10)(8,15)(9,14)(11,18)(12,13)(16,20)(17,19)(21,24,28,25)(22,30,33,47)(23,34,37,40)(26,39,27,41)(29,43,46,52)(31,42,38,32)(35,49,36,51)(44,45,48,50);(2,4)(7,14)(10,13)(15,20)(18,19)(21,25,28,24)(22,31,33,38)(23,34,37,40)(26,39,27,41)(29,44,46,48)(30,42,47,32)(35,51,36,49)(43,45,52,50);(5,8,9,16)(6,11,12,17)(7,15,14,20)(10,18,13,19)(21,26,28,27)(22,32,33,42)(23,35,37,36)(24,39,25,41)(29,45,46,50)(30,31,47,38)(34,49,40,51)(43,48,52,44)Polycyclic presentation:
2^9|a^2=g,b^2=fh,c^2=i,d^2=i,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=ehi,c^b=ch,d^b=dfi,e^b=eh,d^c=dh,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7089788);