Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500456 |
| 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 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1750 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2,3,4)(13,14)(15,20)(16,26)(17,22)(18,32)(19,23)(21,27)(24,36)(25,33)(28,39)(29,34)(30,37)(31,42)(35,40)(38,44)(41,43);(2,4)(5,6)(7,10)(8,9)(11,12)(13,15,19,18)(14,20,23,32)(16,27,29,24)(17,25,30,31)(21,34,36,26)(22,33,37,42)(28,40,41,38)(35,43,44,39);(6,9)(7,11)(13,16,18,27,19,29,15,24)(14,21,20,26,23,36,32,34)(17,28,31,40,30,41,25,38)(22,35,33,39,37,44,42,43);(5,7,8,11)(6,10,9,12)(13,17,19,30)(14,22,23,37)(15,25,18,31)(16,28,29,41)(20,33,32,42)(21,35,36,44)(24,38,27,40)(26,39,34,43);(6,9)(10,12)(17,30)(22,37)(25,31)(28,41)(33,42)(35,44)(38,40)(39,43)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=g,d^2=h,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=ehi,d^c=dhi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6427374);