Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573615 |
| 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×#3013 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2673 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)(7,12)(8,10)(9,11)(13,14)(15,20)(16,27)(17,22)(18,32)(19,23)(21,30)(24,41)(25,35)(26,37)(28,44)(29,40)(31,36)(33,39)(34,43)(38,42);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(13,15,19,26)(14,20,23,37)(16,28,29,24)(17,31,18,25)(21,38,39,34)(22,36,32,35)(27,44,40,41)(30,42,33,43);(2,4)(13,16)(14,21)(15,24)(17,30)(18,33)(19,29)(20,34)(22,40)(23,39)(25,42)(26,28)(27,32)(31,43)(35,41)(36,44)(37,38);(1,2)(3,4)(5,8,9,7)(6,10,11,12)(16,29)(21,39)(24,28)(27,40)(30,33)(34,38)(41,44)(42,43);(2,4)(13,17,19,18)(14,22,23,32)(15,25,26,31)(16,30,29,33)(20,35,37,36)(21,40,39,27)(24,42,28,43)(34,41,38,44)Polycyclic presentation:
2^9|b^2=fi,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=eh,d^c=dh,g^c=gi,e^d=ehi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6484087);