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×D4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2399 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1719 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,3)(5,6,9,8)(7,10,12,11)(13,14)(15,21)(16,30)(17,36)(18,24)(19,25)(20,26)(22,33)(23,29)(27,43)(28,38)(31,40)(32,37)(34,42)(35,41)(39,44);(1,2,4,3)(5,7)(6,10)(8,11)(9,12)(13,15)(14,21)(16,31)(17,19)(18,28)(20,29)(22,39)(23,26)(24,38)(25,36)(27,34)(30,40)(32,35)(33,44)(37,41)(42,43);(5,8,9,6)(7,11,12,10)(13,16,19,34,20,35,18,33)(14,22,24,41,26,42,25,30)(15,27,28,31,29,44,17,32)(21,37,36,39,23,40,38,43);(7,12)(10,11)(13,17,18,29,20,28,19,15)(14,23,25,36,26,21,24,38)(16,32,33,44,35,31,34,27)(22,40,30,39,42,37,41,43);(1,3,4,2)(13,18,20,19)(14,24,26,25)(15,28,29,17)(16,33,35,34)(21,38,23,36)(22,41,42,30)(27,31,44,32)(37,43,40,39)Polycyclic presentation:
2^9|a^2=gh,b^2=f,c^2=h,d^2=g,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, 7511447);