Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12324 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | #447 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ #1770 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
361 conjugacy classes of subgroups; only the 98 normal subgroups are shown.
This group has 361 conjugacy classes of subgroups — too many to draw. Only its 98 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 3 generators on 44 points:
(2,4)(5,6,9,12,10,7,8,11)(13,14)(15,23,25,40)(16,19)(17,29)(18,32)(20,24,34,42)(21,31)(22,28)(26,37,33,41)(27,39)(30,38,44,36)(35,43);(1,2,3,4)(5,7,10,6)(8,11,9,12)(14,19)(17,27)(20,34)(22,32)(24,42)(30,44)(31,43)(36,38);(5,6,10,7)(8,12,9,11)(13,15)(14,20)(16,25)(17,30)(18,33)(19,34)(21,37)(22,38)(23,39)(24,43)(26,28)(27,44)(29,40)(31,42)(32,36)(35,41)Polycyclic presentation:
2^9|a^2=f,b^2=dfgi,c^2=g,d^2=g,e^2=gh,f^2=g,h^2=i,b^a=bd,c^a=ce,e^a=eh,h^a=hi,d^b=dg,e^b=egi,f^b=fg,d^c=dgi,e^c=eghi,f^c=fgi,h^c=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 44767);