Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6629011 |
| 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 | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2236 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)(7,9)(8,10)(11,12)(13,14,20,26)(15,27,29,39)(16,22,32,40)(17,35,37,43)(18,25,19,24)(21,31,38,30)(23,36,44,28)(33,42,34,41);(5,7,8,11)(6,9,10,12)(13,15,19,31,20,29,18,30)(14,21,25,39,26,38,24,27)(16,28,34,37,32,36,33,17)(22,35,42,44,40,43,41,23);(6,10)(9,12)(13,16)(14,22)(15,28)(17,30)(18,33)(19,34)(20,32)(21,35)(23,27)(24,41)(25,42)(26,40)(29,36)(31,37)(38,43)(39,44);(2,4)(13,17,18,36,20,37,19,28)(14,23,24,43,26,44,25,35)(15,16,30,33,29,32,31,34)(21,22,27,41,38,40,39,42);(15,29)(16,32)(21,38)(22,40)(27,39)(30,31)(33,34)(41,42)Polycyclic presentation:
2^9|a^2=i,b^2=fgi,d^2=f,f^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,h^a=hi,e^b=ei,e^c=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6536948);