Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #151256 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2196 |
| commutator, Frattini, cosocle | #390 | C23 |
| prime radical | #30551 | C2 |
| LCS | G ▷ #390 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ #390 ◁ G | |
| LECS | G ▷ #390 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
395 conjugacy classes of subgroups; only the 38 normal subgroups are shown.
This group has 395 conjugacy classes of subgroups — too many to draw. Only its 38 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:
(2,7,10,24)(3,12)(4,17)(8,14,20,31)(9,29,11,23)(13,16)(15,32)(18,26)(19,30)(22,28,25,27);(2,8)(3,13)(4,18)(7,22)(9,28)(10,20)(11,27)(12,30)(14,29)(15,19)(16,32)(17,26)(23,31)(24,25);(1,2,4,9,6,10,18,11)(3,14,16,29,12,31,19,23)(5,20,17,28,21,8,26,27)(7,13,22,15,24,30,25,32)Polycyclic presentation:
2^9|a^2=f,c^2=fg,d^2=f,g^2=i,b^a=bd,c^a=ce,d^a=df,e^a=ei,g^a=gh,c^b=ch,d^b=df,e^b=eghi,g^b=gi,d^c=dg,e^c=eh,f^c=fi,g^c=gi,h^c=hi,e^d=eh,g^e=gi35 conjugacy classes — view the full character table.
| maximal subgroup of | #72339074 |
| maximal normal subgroup of | #72339074 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 58664);