Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #93229 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #353 |
| commutator | C2×#97 | C2×C4 |
| Frattini, cosocle | #1985 | C22 |
| socle | C22 | #1343 |
| prime radical | #3150 | C4 |
| LCS | G ▷ C2×#97 ▷ C2×C42 ▷ C23 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C42 ◁ #1985 ◁ G | |
| LECS | G ▷ #1985 ▷ C2×C42 ▷ C23 ▷ C22 ▷ C1 | |
189 conjugacy classes of subgroups; only the 26 normal subgroups are shown.
This group has 189 conjugacy classes of subgroups — too many to draw. Only its 26 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 2 generators on 48 points:
(1,2,3,8,5,9,11,16)(4,7,10,15,13,6,12,14)(17,18,21,26,23,28,22,27)(19,29,31,45,33,46,32,42)(20,34,36,43,37,44,38,25)(24,35,39,30,41,47,40,48);(2,6)(3,10)(4,13)(7,9)(11,12)(14,15)(17,19)(18,24)(20,35)(21,32)(22,31)(23,33)(25,42)(26,40)(27,39)(28,41)(29,34)(30,36)(37,47)(38,48)(43,45)(44,46)Polycyclic presentation:
2^9|a^2=d,c^2=fi,d^2=hi,e^2=hi,f^2=i,b^a=bc,c^a=ce,e^a=eg,f^a=fh,g^a=gh,c^b=cf,d^b=defhi,e^b=ehi,f^b=fi,g^b=gi,d^c=dghi,e^c=ei,e^d=eh44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 431);