Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #93862 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1361 |
| commutator | C4×C16 | C2×C4 |
| Frattini, cosocle | C8×C16 | C22 |
| prime radical | #2246 | C4 |
| LCS | G ▷ C4×C16 ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ C82 ◁ C8×C16 ◁ G | |
| LECS | G ▷ C8×C16 ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
104 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 2 generators on 48 points:
(1,2,3,8,11,6,10,14,5,9,13,16,4,7,12,15)(17,18)(19,23)(20,32)(21,35)(22,27)(24,28)(25,29)(26,33)(30,42)(31,38)(34,48)(36,44)(37,43)(39,47)(40,45)(41,46);(2,6)(3,10)(4,11)(7,9)(8,16)(12,13)(18,23)(19,28)(20,29)(21,33)(24,25)(26,42)(27,38)(30,47)(31,45)(32,36)(34,46)(35,37)(39,48)(40,41)(43,44)Polycyclic presentation:
2^9|a^2=d,c^2=fh,d^2=efg,e^2=gi,f^2=hi,g^2=i,h^2=i,b^a=bc,c^a=ce,e^a=eg,f^a=fg,g^a=gi,h^a=hi,c^b=cf,d^b=defg,e^b=eg,f^b=fh,g^b=gi,h^b=hi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1058);