Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of D4×AGL(1,ℤ/8) |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C42 | C25 |
| socle | C22 | C22×C4×D4 |
| prime radical | #61447 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C8×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1544 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,9,6,3,7,11)(4,8,13,15,12,10,14,16)(17,18)(19,28)(20,32)(21,27)(22,36)(23,25)(24,30)(26,34)(29,44)(31,40)(33,42)(35,41)(37,38)(39,45)(43,47)(46,48);(17,19)(18,24)(20,29)(21,30)(22,37)(23,31)(25,38)(26,39)(27,40)(28,36)(32,48)(33,45)(34,47)(35,46)(41,43)(42,44);(1,3,5,11,6,2,7,9)(4,10,13,16,12,8,14,15)(18,25)(24,38)(26,32)(27,36)(28,40)(39,48)(41,42)(43,44);(1,4)(2,8)(3,10)(5,13)(6,12)(7,14)(9,15)(11,16)(17,20,23,34)(18,26,25,32)(19,29,31,47)(21,35,22,33)(24,39,38,48)(27,42,36,41)(28,43,40,44)(30,46,37,45);(4,12)(8,10)(13,14)(15,16)(17,21,23,22)(18,27,25,36)(19,30,31,37)(20,33,34,35)(24,40,38,28)(26,41,32,42)(29,45,47,46)(39,43,48,44)Polycyclic presentation:
2^9|a^2=g,c^2=g,d^2=h,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=ch,d^a=dh,f^a=fh,f^b=fh,e^d=ehi110 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6253072);