Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| 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 | |
2052 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)(13,14,17,22,18,20,28,33)(15,19,25,30,23,31,36,41)(16,21,27,34,26,32,38,42)(24,29,37,40,35,39,43,44);(5,7,8,12)(6,9,10,11)(16,26)(21,32)(24,35)(27,38)(29,39)(34,42)(37,43)(40,44);(2,4)(7,12)(9,11)(13,15)(14,19)(16,24)(17,25)(18,23)(20,31)(21,29)(22,30)(26,35)(27,37)(28,36)(32,39)(33,41)(34,40)(38,43)(42,44);(6,10)(9,11)(14,20)(15,23)(16,26)(22,33)(25,36)(27,38)(29,39)(40,44);(13,16,17,27,18,26,28,38)(14,21,22,34,20,32,33,42)(15,24,25,37,23,35,36,43)(19,29,30,40,31,39,41,44)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,c^b=cf,e^b=ei,d^c=di,e^d=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6284893);