Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6433044 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #35328 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2038 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)(7,11)(10,12)(13,14)(15,23)(16,20)(17,30)(18,21)(19,25)(22,31)(24,37)(26,38)(27,42)(28,35)(29,44)(32,40)(33,41)(34,43)(36,39);(5,6)(7,10)(8,9)(11,12)(13,15,17,25,18,26,31,41)(14,19,22,23,21,33,30,38)(16,27,29,43,28,24,39,40)(20,34,36,42,35,32,44,37);(2,4)(6,9)(10,12)(13,16)(14,20)(15,24)(17,29)(18,28)(19,32)(21,35)(22,36)(23,37)(25,40)(26,27)(30,44)(31,39)(33,34)(38,42)(41,43);(5,7,8,11)(6,10,9,12)(14,21)(19,33)(20,35)(22,30)(23,38)(32,34)(36,44)(37,42);(16,28)(20,35)(24,27)(29,39)(32,34)(36,44)(37,42)(40,43)Polycyclic presentation:
2^9|a^2=g,b^2=f,d^2=hi,f^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,c^b=ch,e^c=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6342749);