Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573615 |
| 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×#3524 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3142 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)(13,14,19,24)(15,25,27,36)(16,30,33,37)(17,23,18,22)(20,28,35,29)(21,32,38,31)(26,40,41,34)(39,42,44,43);(2,4)(5,6,8,9)(7,10,11,12)(13,15,19,27)(14,20,24,35)(16,26,33,41)(17,28,18,29)(21,34,38,40)(22,36,23,25)(30,44,37,39)(31,42,32,43);(6,9)(7,11)(13,16)(14,21)(15,26)(17,31)(18,32)(19,33)(20,34)(22,30)(23,37)(24,38)(25,39)(27,41)(28,42)(29,43)(35,40)(36,44);(5,7)(6,10)(8,11)(9,12)(15,27)(20,35)(25,36)(26,41)(28,29)(34,40)(39,44)(42,43);(6,9)(7,11)(13,17,19,18)(14,22,24,23)(15,28,27,29)(16,31,33,32)(20,36,35,25)(21,30,38,37)(26,42,41,43)(34,44,40,39)Polycyclic presentation:
2^9|a^2=i,b^2=hi,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,g^a=gi,c^b=ch,d^b=di,e^b=eh,f^b=fi,g^b=gi,d^c=dh,f^c=fi,g^c=gi,e^d=eh77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6503189);