Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #93770 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1343 |
| commutator | C2×C22⋊C8 | C2×C4 |
| Frattini, cosocle | #1817 | C22 |
| prime radical | #3186 | C4 |
| LCS | G ▷ C2×C22⋊C8 ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ #1817 ◁ G | |
| LECS | G ▷ #1817 ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1 | |
238 conjugacy classes of subgroups; only the 30 normal subgroups are shown.
This group has 238 conjugacy classes of subgroups — too many to draw. Only its 30 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) = 128 |
A minimal faithful permutation representation — 2 generators on 48 points:
(1,2)(3,11,14,6)(4,8)(5,10)(7,12,15,16)(9,13)(17,18,21,25,22,26,34,42)(19,27,29,43,30,39,46,48)(20,31,32,47,33,24,38,40)(23,35,36,28,37,44,41,45);(2,6)(3,12)(4,13)(7,9)(8,15)(10,11)(14,16)(17,19)(18,23)(20,28)(21,29)(22,30)(24,39)(25,41)(26,37)(27,31)(32,35)(33,45)(34,46)(36,42)(38,44)(40,48)(43,47)Polycyclic presentation:
2^9|a^2=d,c^2=fh,d^2=i,e^2=h,f^2=h,b^a=bc,c^a=ce,e^a=eg,f^a=fhi,c^b=cf,d^b=defh,e^b=eh,f^b=fh,g^b=gi,d^c=dgh,e^c=ei44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 967);