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 | C22×2−1+4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3044 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)(7,9)(8,10)(11,12)(13,14,20,26)(15,22,29,35)(16,21,28,36)(17,23,31,38)(18,25,19,24)(27,37,41,44)(30,34,42,43)(32,40,33,39);(1,2,3,4)(5,7)(6,9)(8,11)(10,12)(13,15,19,28,20,29,18,16)(14,21,25,35,26,36,24,22)(17,27,33,42,31,41,32,30)(23,34,40,44,38,43,39,37);(6,10)(7,11)(13,16,18,29,20,28,19,15)(14,22,24,36,26,35,25,21)(17,30,32,41,31,42,33,27)(23,37,39,43,38,44,40,34);(7,11)(9,12)(13,17)(14,23)(15,27)(16,30)(18,32)(19,33)(20,31)(21,34)(22,37)(24,39)(25,40)(26,38)(28,42)(29,41)(35,44)(36,43);(17,31)(23,38)(27,41)(30,42)(32,33)(34,43)(37,44)(39,40)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=ghi,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=dh,e^d=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6366973);