Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806839 |
| 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 | #42926 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2122 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,25)(16,29)(17,23)(18,22)(19,24)(20,27)(21,28)(26,39)(30,35)(31,42)(32,37)(33,38)(34,40)(36,41)(43,44);(6,9)(7,11)(13,15,18,27)(14,20,22,25)(16,30,28,19)(17,31,32,41)(21,35,29,24)(23,36,37,42)(26,33,40,43)(34,38,39,44);(1,2,3,4)(13,16,19,15,18,28,30,27)(14,21,24,20,22,29,35,25)(17,31,33,26,32,41,43,40)(23,36,38,34,37,42,44,39);(6,9)(10,12)(14,22)(17,32)(20,25)(21,29)(24,35)(26,40)(31,41)(33,43);(5,7,8,11)(6,10,9,12)(13,17,30,43,18,32,19,33)(14,23,35,44,22,37,24,38)(15,26,16,31,27,40,28,41)(20,34,21,36,25,39,29,42)Polycyclic presentation:
2^9|b^2=i,c^2=ghi,e^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,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, 6723340);