Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | #36687 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2052 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)(5,6)(7,11)(8,10)(9,12)(13,14,17,21,18,22,29,35)(15,19,25,32,23,30,37,41)(16,26,28,40,27,20,33,34)(24,38,39,44,36,31,42,43);(5,7,8,12)(6,9,10,11)(15,23)(19,30)(24,36)(25,37)(31,38)(32,41)(39,42)(43,44);(2,4)(7,12)(9,11)(13,15)(14,19)(16,24)(17,25)(18,23)(20,31)(21,32)(22,30)(26,38)(27,36)(28,39)(29,37)(33,42)(34,43)(35,41)(40,44);(6,10)(9,11)(15,23)(16,27)(19,30)(20,26)(25,37)(28,33)(32,41)(34,40);(13,16,17,28,18,27,29,33)(14,20,21,34,22,26,35,40)(15,24,25,39,23,36,37,42)(19,31,32,43,30,38,41,44)Polycyclic presentation:
2^9|a^2=h,b^2=f,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=cfi,d^c=di,e^d=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6284983);