Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61591 |
| 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 | |
1519 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,3,6,8,13,10)(4,7,12,9,11,14,16,15)(17,18,23,27)(19,28,31,39)(20,32,34,25)(21,26,22,36)(24,37,40,30)(29,44,47,43)(33,42,35,41)(38,46,48,45);(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);(1,3,5,8,6,10,13,2)(4,9,12,14,11,15,16,7);(1,4)(2,7)(3,9)(5,12)(6,11)(8,14)(10,15)(13,16)(17,20,23,34)(18,25,27,32)(19,29,31,47)(21,35,22,33)(24,38,40,48)(26,42,36,41)(28,43,39,44)(30,46,37,45);(4,11)(7,14)(9,15)(12,16)(17,21,23,22)(18,26,27,36)(19,30,31,37)(20,33,34,35)(24,39,40,28)(25,41,32,42)(29,45,47,46)(38,43,48,44)Polycyclic presentation:
2^9|a^2=g,c^2=gh,d^2=h,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=ci,d^a=dh,f^a=fh,f^b=fh,e^d=ehi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6253366);