Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667932 |
| 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 | #2414 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1855 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)(7,10)(8,12)(9,11)(13,14,19,23)(15,24,25,35)(16,21,29,38)(17,22,18,31)(20,32,33,27)(26,39,40,44)(28,36,30,37)(34,41,43,42);(1,2)(3,4)(5,7,10,6)(8,12,9,11)(15,25)(20,33)(24,35)(26,40)(27,32)(34,43)(39,44)(41,42);(2,4)(13,15,19,25)(14,20,23,33)(16,26,29,40)(17,27,18,32)(21,34,38,43)(22,35,31,24)(28,41,30,42)(36,39,37,44);(5,8,10,9)(6,11,7,12)(13,16)(14,21)(15,26)(17,30)(18,28)(19,29)(20,34)(22,37)(23,38)(24,39)(25,40)(27,42)(31,36)(32,41)(33,43)(35,44);(2,4)(5,9,10,8)(6,11,7,12)(13,17,19,18)(14,22,23,31)(15,27,25,32)(16,28,29,30)(20,35,33,24)(21,36,38,37)(26,41,40,42)(34,39,43,44)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=i,d^2=f,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efhi,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6584221);