Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518944 |
| 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×D8 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2399 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1823 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)(15,23)(16,27)(17,31)(18,22)(19,33)(20,25)(21,30)(24,37)(26,36)(28,35)(29,43)(32,34)(38,42)(39,41)(40,44);(2,4)(6,9)(10,11)(13,15,18,25)(14,20,22,23)(16,28,29,41)(17,32,33,36)(19,34,31,26)(21,37,38,44)(24,30,40,42)(27,39,43,35);(13,16,19,30,18,29,31,42)(14,21,17,27,22,38,33,43)(15,24,26,41,25,40,34,28)(20,35,36,44,23,39,32,37);(1,2,3,4)(13,17,31,14,18,33,19,22)(15,23,34,36,25,20,26,32)(16,27,42,21,29,43,30,38)(24,39,28,44,40,35,41,37);(5,7,8,12)(6,10,9,11)(14,22)(17,33)(20,23)(21,38)(27,43)(32,36)(35,39)(37,44)Polycyclic presentation:
2^9|b^2=i,c^2=g,d^2=hi,e^2=fi,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=cgi,d^b=dh,g^b=gi,h^b=hi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7521147);