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 | C2×#550 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1851 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)(8,12)(9,10)(15,25)(16,30)(17,18)(20,26)(21,38)(22,23)(27,32)(28,29)(31,40)(33,34)(35,39)(36,37);(2,4)(6,10)(11,12)(13,14,19,24)(15,26,29,36)(16,31,33,38)(17,23,18,22)(20,28,37,25)(21,30,40,34)(27,35,32,39)(41,44,43,42);(5,7,9,8)(6,11,10,12)(13,15,18,28,19,29,17,25)(14,20,23,36,24,37,22,26)(16,32,34,41,33,27,30,43)(21,39,31,42,40,35,38,44);(1,2,3,4)(13,16,17,30,19,33,18,34)(14,21,22,38,24,40,23,31)(15,27,25,41,29,32,28,43)(20,35,26,42,37,39,36,44);(5,8,9,7)(6,11,10,12)(16,33)(21,40)(27,32)(30,34)(31,38)(35,39)(41,43)(42,44)Polycyclic presentation:
2^9|b^2=i,c^2=fgi,d^2=hi,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, 7521629);