Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993947 |
| 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 | #42908 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1892 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2,3,4)(5,6)(7,9)(8,10)(11,12)(13,14)(15,25)(16,22)(17,23)(18,34)(19,24)(20,28)(21,31)(26,36)(27,41)(29,37)(30,40)(32,38)(33,39)(35,43)(42,44);(5,6)(7,11)(8,10)(9,12)(13,15,18,28,19,29,17,32)(14,20,23,25,24,38,34,37)(16,26,30,27,31,44,33,43)(21,35,40,42,22,41,39,36);(2,4)(5,7)(6,9)(8,12)(10,11)(13,16)(14,21)(15,26)(17,33)(18,30)(19,31)(20,35)(22,24)(23,40)(25,42)(27,28)(29,44)(32,43)(34,39)(36,37)(38,41);(7,12)(9,11)(13,16)(14,22)(15,27)(17,30)(18,33)(19,31)(20,36)(21,24)(23,40)(25,41)(26,28)(29,43)(32,44)(34,39)(35,37)(38,42);(6,10)(9,11)(13,17,19,18)(14,23,24,34)(15,28,29,32)(16,30,31,33)(20,37,38,25)(21,39,22,40)(26,43,44,27)(35,42,41,36)Polycyclic presentation:
2^9|a^2=gi,b^2=f,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dfi,e^b=ehi,d^c=dh,e^c=ei,f^d=fi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6910756);