Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573591 |
| 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×#3489 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4122 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2)(3,4)(5,6,8,10)(7,9,11,12)(13,14)(15,24)(16,21)(17,22)(18,31)(19,23)(20,26)(25,38)(27,34)(28,36)(29,37)(30,41)(32,35)(33,40)(39,44)(42,43);(5,7)(6,9)(8,11)(10,12)(13,15,19,27)(14,20,23,35)(16,25,28,39)(17,26,18,32)(21,33,36,43)(22,34,31,24)(29,40,30,42)(37,44,41,38);(2,4)(7,11)(9,12)(13,16,19,28)(14,21,23,36)(15,25,27,39)(17,29,18,30)(20,33,35,43)(22,37,31,41)(24,38,34,44)(26,40,32,42);(5,7)(6,9)(8,11)(10,12)(16,28)(21,36)(25,39)(29,30)(33,43)(37,41)(38,44)(40,42);(7,11)(9,12)(13,17,19,18)(14,22,23,31)(15,26,27,32)(16,29,28,30)(20,34,35,24)(21,37,36,41)(25,40,39,42)(33,44,43,38)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,f^b=fi,d^c=dhi,e^d=eh83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6498729);