Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437521 |
| 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 | C22×#135 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2285 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2162 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)(7,11)(8,9)(10,12)(13,14)(15,25)(16,22)(17,32)(18,24)(19,23)(20,28)(21,31)(26,35)(27,37)(29,39)(30,40)(33,38)(34,41)(36,44)(42,43);(2,4)(6,9)(10,11)(13,15)(14,20)(16,29)(17,27)(18,28)(19,33)(21,39)(22,42)(23,37)(24,25)(26,30)(31,43)(32,38)(34,44)(35,41)(36,40);(5,7,8,12)(6,10,9,11)(13,16)(14,21)(15,26)(17,30)(18,31)(19,34)(20,35)(22,24)(23,40)(25,36)(27,43)(28,44)(29,33)(32,41)(37,39)(38,42);(13,16)(14,22)(15,26)(17,30)(18,31)(19,34)(20,36)(21,24)(23,41)(25,35)(27,43)(28,44)(29,33)(32,40)(37,42)(38,39);(13,17,18,19)(14,23,24,32)(15,27,28,33)(16,30,31,34)(20,37,25,38)(21,40,22,41)(26,43,44,29)(35,39,36,42)Polycyclic presentation:
2^9|a^2=fg,c^2=gi,e^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,h^b=hi,h^c=hi,h^d=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6352755);