Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514040 |
| 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 | #45245 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2035 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)(13,14,19,23)(15,20,27,35)(16,21,28,36)(17,30,18,22)(24,32,41,43)(25,33,26,34)(29,38,31,37)(39,42,40,44);(1,2)(3,4)(6,9)(10,12)(13,15)(14,20)(16,24)(17,26)(18,25)(19,27)(21,32)(22,33)(23,35)(28,41)(29,40)(30,34)(31,39)(36,43)(37,42)(38,44);(2,4)(13,16,19,28)(14,21,23,36)(15,24,27,41)(17,31,18,29)(20,32,35,43)(22,38,30,37)(25,40,26,39)(33,44,34,42);(2,4)(5,7,8,11)(6,10,9,12)(15,25)(16,28)(17,18)(20,33)(21,36)(22,30)(24,39)(26,27)(32,42)(34,35)(40,41)(43,44);(6,9)(10,12)(13,17,19,18)(14,22,23,30)(15,26,27,25)(16,29,28,31)(20,33,35,34)(21,37,36,38)(24,40,41,39)(32,42,43,44)Polycyclic presentation:
2^9|a^2=i,c^2=i,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^c=ei,e^d=ei,h^d=hi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7423685);