Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6469971 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3177 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2)(3,4)(7,11)(10,12)(13,14)(15,24)(16,21)(17,22)(18,33)(19,23)(20,27)(25,40)(26,35)(28,39)(29,43)(30,37)(31,38)(32,44)(34,42)(36,41);(5,6)(7,10)(8,9)(11,12)(13,15,19,28)(14,20,23,35)(16,29,31,25)(17,27,18,26)(21,36,38,34)(22,39,33,24)(30,41,32,42)(37,40,44,43);(2,4)(6,9)(10,12)(13,16)(14,21)(15,25)(17,30)(18,32)(19,31)(20,34)(22,37)(23,38)(24,40)(26,41)(27,42)(28,29)(33,44)(35,36)(39,43);(6,9)(10,12)(13,17,19,18)(14,22,23,33)(15,26,28,27)(16,30,31,32)(20,24,35,39)(21,37,38,44)(25,41,29,42)(34,40,36,43);(5,7,8,11)(6,10,9,12)(14,23)(16,31)(20,35)(22,33)(24,39)(25,29)(30,32)(41,42)Polycyclic presentation:
2^9|b^2=i,d^2=i,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,f^b=fi,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6398073);