Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437758 |
| 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 | #24390 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3088 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6)(7,10)(8,9)(11,12)(13,14)(15,25)(16,21)(17,22)(18,33)(19,24)(20,28)(23,30)(26,39)(27,34)(29,40)(31,37)(32,38)(35,42)(36,41)(43,44);(1,2)(3,4)(13,15,18,28,19,29,30,41)(14,20,23,25,24,36,33,40)(16,26,17,27,31,42,32,43)(21,34,38,39,37,44,22,35);(2,4)(6,9)(10,12)(13,16)(14,21)(15,26)(17,18)(19,31)(20,34)(22,33)(23,38)(24,37)(25,39)(27,28)(29,42)(30,32)(35,40)(36,44)(41,43);(2,4)(13,17)(14,22)(15,27)(16,30)(18,31)(19,32)(20,35)(21,23)(24,38)(25,34)(26,41)(28,42)(29,43)(33,37)(36,39)(40,44);(5,7,8,11)(6,10,9,12)(16,31)(17,32)(21,37)(22,38)(26,42)(27,43)(34,44)(35,39)Polycyclic presentation:
2^9|b^2=f,e^2=g,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,d^c=di,e^c=ei,e^d=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6361835);