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 | |
1716 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)(13,14,19,24)(15,26,28,21)(16,30,34,40)(17,36,29,38)(18,25,20,23)(22,33,41,35)(27,39,43,44)(31,37,32,42);(1,2,4,3)(5,7)(6,10)(8,11)(9,12)(13,15)(14,21)(16,31)(17,18)(19,28)(20,29)(22,39)(23,36)(24,26)(25,38)(27,33)(30,42)(32,34)(35,43)(37,40)(41,44);(5,8,9,6)(7,11,12,10)(13,16,18,33,19,34,20,35)(14,22,23,40,24,41,25,30)(15,27,29,31,28,43,17,32)(21,37,38,39,26,42,36,44);(7,12)(10,11)(13,17,20,28,19,29,18,15)(14,21,25,36,24,26,23,38)(16,32,35,43,34,31,33,27)(22,37,30,44,41,42,40,39);(1,3,4,2)(13,18,19,20)(14,23,24,25)(15,17,28,29)(16,33,34,35)(21,36,26,38)(22,40,41,30)(27,32,43,31)(37,44,42,39)Polycyclic presentation:
2^9|a^2=ghi,b^2=fi,c^2=h,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=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, 7511444);