Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519008 |
| 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×#135 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2150 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6)(7,11)(8,9)(10,12)(13,14,19,24)(15,26,28,21)(16,29,33,40)(17,35,36,42)(18,25,20,23)(22,32,41,34)(27,43,44,38)(30,39,31,37);(1,2)(3,4)(6,9)(10,11)(13,15,19,28)(14,21,24,26)(16,30,33,31)(17,20,36,18)(22,38,41,43)(23,35,25,42)(27,32,44,34)(29,37,40,39);(13,16,18,32,19,33,20,34)(14,22,23,40,24,41,25,29)(15,27,36,31,28,44,17,30)(21,37,42,43,26,39,35,38);(2,4)(13,17,20,28,19,36,18,15)(14,21,25,35,24,26,23,42)(16,31,34,27,33,30,32,44)(22,39,29,43,41,37,40,38);(5,7,8,12)(6,10,9,11)(13,18,19,20)(14,23,24,25)(15,17,28,36)(16,32,33,34)(21,35,26,42)(22,40,41,29)(27,30,44,31)(37,38,39,43)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=gi,d^2=g,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,d^c=di,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7469716);