Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438975 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2051 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(5,6)(7,11)(8,12)(9,10)(13,14,20,25,19,24,18,23)(15,26,17,21,29,36,28,37)(16,30,34,39,33,22,32,40)(27,38,31,44,43,41,42,35);(6,10)(11,12)(13,15)(14,21)(16,31)(17,18)(19,29)(20,28)(22,38)(23,36)(24,37)(25,26)(27,34)(30,41)(32,43)(33,42)(35,40)(39,44);(1,2,3,4)(5,7,9,8)(6,11,10,12)(13,16,20,34,19,33,18,32)(14,22,25,40,24,30,23,39)(15,27,17,31,29,43,28,42)(21,35,36,38,37,44,26,41);(5,8,9,7)(6,12,10,11)(13,17,18,15,19,28,20,29)(14,21,23,26,24,37,25,36)(16,31,32,27,33,42,34,43)(22,35,39,41,30,44,40,38);(13,18,19,20)(14,23,24,25)(15,17,29,28)(16,32,33,34)(21,36,37,26)(22,39,30,40)(27,31,43,42)(35,38,44,41)Polycyclic presentation:
2^9|a^2=h,c^2=fg,d^2=f,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^d=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6364917);