Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | C2×#121 | C2×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1778 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)(6,9)(7,11)(13,14,17,21,18,22,29,36)(15,19,25,32,24,31,38,42)(16,20,28,35,27,34,40,44)(23,30,37,41,26,33,39,43);(5,6)(7,10)(8,9)(11,12)(13,15)(14,19)(16,26)(17,25)(18,24)(20,33)(21,32)(22,31)(23,27)(28,39)(29,38)(30,34)(35,43)(36,42)(37,40)(41,44);(2,4)(13,16,17,28,18,27,29,40)(14,20,21,35,22,34,36,44)(15,23,25,37,24,26,38,39)(19,30,32,41,31,33,42,43);(6,9)(10,12)(16,27)(20,34)(23,26)(28,40)(30,33)(35,44)(37,39)(41,43);(5,7,8,11)(6,10,9,12)(15,24)(16,27)(19,31)(20,34)(25,38)(28,40)(32,42)(35,44)Polycyclic presentation:
2^9|a^2=h,c^2=h,e^2=f,h^2=i,b^a=bf,c^a=cg,e^a=ef,c^b=ci,d^b=df,e^b=ei,d^c=di,e^c=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6300011);