Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37436 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | #36687 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2060 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)(8,10)(9,12)(13,14,17,22,18,20,30,34)(15,19,25,33,23,31,38,42)(16,26,28,40,29,21,35,37)(24,36,39,44,27,32,41,43);(5,7,8,12)(6,9,10,11)(15,23)(19,31)(24,27)(25,38)(32,36)(33,42)(39,41)(43,44);(2,4)(7,12)(9,11)(13,15)(14,19)(16,27)(17,25)(18,23)(20,31)(21,36)(22,33)(24,29)(26,32)(28,41)(30,38)(34,42)(35,39)(37,44)(40,43);(6,10)(9,11)(14,20)(15,23)(21,26)(22,34)(24,27)(25,38)(37,40)(39,41);(13,16,17,28,18,29,30,35)(14,21,22,37,20,26,34,40)(15,24,25,39,23,27,38,41)(19,32,33,43,31,36,42,44)Polycyclic presentation:
2^9|a^2=h,b^2=f,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=dfi,e^a=ei,c^b=cfi,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, 6284903);