Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #152610 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13049 |
| commutator | #124 | C22×C4 |
| Frattini, cosocle | C2×#124 | C23 |
| prime radical | #18992 | C2 |
| LCS | G ▷ #124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C22×C4 ◁ C2×#124 ◁ G | |
| LECS | G ▷ C2×#124 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
494 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 494 conjugacy classes of subgroups — too many to draw. Only its 60 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,11,14,30)(4,16,18,17)(5,9,20,27)(6,10)(7,13,22,31)(8,19)(12,29,15,25)(21,28,23,32)(24,26);(1,2)(3,7,15,23)(4,17)(5,9)(6,10)(8,24)(11,28,25,31)(12,21,14,22)(13,30,32,29)(16,18)(19,26)(20,27);(1,3)(2,7)(4,13)(5,12)(6,15)(8,25)(9,21)(10,23)(11,26)(14,20)(16,29)(17,30)(18,32)(19,31)(22,27)(24,28)Polycyclic presentation:
2^9|a^2=f,b^2=df,e^2=hi,g^2=i,b^a=bd,c^a=ce,e^a=eghi,g^a=gh,h^a=hi,c^b=ch,e^b=egh,h^b=hi,d^c=dg,e^c=ehi,f^c=fgi,g^c=gi,e^d=eh,g^d=gi,f^e=fh,g^f=gi35 conjugacy classes — view the full character table.
| maximal subgroup of | #72219521, #72219528, #72219529, … |
| maximal normal subgroup of | #72219521, #72219528, #72219529, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59806);