Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682440 |
| 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×#3203 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1887 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6)(7,11)(8,10)(9,12)(13,14)(15,20)(16,26)(17,22)(18,33)(19,23)(21,29)(24,36)(25,34)(27,42)(28,43)(30,37)(31,44)(32,39)(35,41)(38,40);(2,3)(5,7)(6,9)(8,12)(10,11)(13,15,19,18)(14,20,23,33)(16,24,28,29)(17,30,32,25)(21,26,36,43)(22,37,39,34)(27,41,31,40)(35,44,38,42);(1,2,4,3)(13,16,18,29,19,28,15,24)(14,21,20,26,23,36,33,43)(17,31,30,40,32,27,25,41)(22,38,34,44,39,35,37,42);(6,10)(9,11)(13,17,19,32)(14,22,23,39)(15,25,18,30)(16,27,28,31)(20,34,33,37)(21,35,36,38)(24,40,29,41)(26,42,43,44);(1,3,4,2)(7,12)(9,11)(16,28)(17,32)(21,36)(22,39)(24,29)(25,30)(26,43)(34,37)Polycyclic presentation:
2^9|b^2=i,c^2=gh,d^2=i,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=di,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6721981);