Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #36702 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | #34641 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2498 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(5,6)(7,10)(8,11)(9,12)(13,14,17,22,18,21,28,34)(15,19,25,31,24,30,37,41)(16,20,27,33,26,32,38,42)(23,29,35,40,36,39,43,44);(5,7,10,6)(8,11,9,12)(13,15,17,25,18,24,28,37)(14,19,22,31,21,30,34,41)(16,23,27,35,26,36,38,43)(20,29,33,40,32,39,42,44);(1,2,3,4)(5,8,10,9)(6,11,7,12)(13,16)(14,20)(15,23)(17,27)(18,26)(19,29)(21,32)(22,33)(24,36)(25,35)(28,38)(30,39)(31,40)(34,42)(37,43)(41,44);(5,9,10,8)(6,11,7,12)(15,24)(16,26)(19,30)(20,32)(25,37)(27,38)(31,41)(33,42);(14,21)(19,30)(20,32)(22,34)(29,39)(31,41)(33,42)(40,44)Polycyclic presentation:
2^9|a^2=h,b^2=fh,c^2=fg,d^2=f,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=cf,d^b=di,d^c=di104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6283601);