Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12310 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1619 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C2×#493 | C22 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
472 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 472 conjugacy classes of subgroups — too many to draw. Only its 90 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,4)(5,6,9,13,25,15,33,32,11,14,26,23,10,7,16,18)(8,19,20,35,27,30,36,31,22,12,28,29,21,17,24,34);(1,2,3,4)(5,7)(6,10)(8,17)(9,23)(11,15)(12,27)(13,26)(14,25)(16,18)(19,21)(20,29)(22,30)(24,34)(28,35)(31,36)(32,33);(5,8,11,22)(6,12,14,19)(7,17,15,30)(9,24,26,36)(10,27,25,21)(13,31,23,34)(16,20,33,28)(18,29,32,35)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,c^2=i,d^2=gi,e^2=gh,f^2=gi,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eh,h^a=hi,d^b=dg,e^b=eg,f^b=fg,g^b=gi,d^c=dg,e^c=egh,f^c=fg,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72224802, #72224809, #72224810, … |
| maximal normal subgroup of | #72224802, #72224809, #72224810, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 44748);