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 | |
1996 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)(7,10)(8,9)(11,12)(13,14)(15,25)(16,30)(17,22)(18,35)(19,24)(20,23)(21,29)(26,37)(27,43)(28,38)(31,40)(32,41)(33,44)(34,42)(36,39);(2,4)(6,9)(10,12)(13,15)(14,21)(16,31)(17,33)(18,28)(19,29)(20,36)(22,40)(23,38)(24,25)(26,32)(27,34)(30,44)(35,39)(37,42)(41,43);(6,9)(10,12)(13,16,19,17)(14,22,24,30)(15,26,29,27)(18,32,20,34)(21,37,25,43)(23,41,35,42)(28,33,36,31)(38,40,39,44);(5,7,8,11)(6,10,9,12)(13,17,19,16)(14,22,24,30)(15,27,29,26)(18,34,20,32)(21,37,25,43)(23,41,35,42)(28,31,36,33)(38,40,39,44);(13,18,19,20)(14,23,24,35)(15,28,29,36)(16,32,17,34)(21,38,25,39)(22,41,30,42)(26,33,27,31)(37,40,43,44)Polycyclic presentation:
2^9|a^2=fg,c^2=i,d^2=g,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, 6357031);