Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #150129 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #11654 |
| commutator | C2×C16 | C22×C4 |
| Frattini, cosocle | #346 | C23 |
| prime radical | #33748 | C2 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×C4×C8 ◁ C4×M32 ◁ G | |
| LECS | G ▷ #346 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
212 conjugacy classes of subgroups; only the 62 normal subgroups are shown.
This group has 212 conjugacy classes of subgroups — too many to draw. Only its 62 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:
(1,2,4,11,6,13,18,31,8,14,23,12,7,25,5,24)(3,15,16,29,20,9,19,27,17,28,22,30,21,26,32,10);(2,9)(3,16)(4,5)(6,7)(10,24)(11,30)(12,29)(13,15)(14,26)(17,22)(18,23)(19,21)(20,32)(25,28)(27,31);(3,17)(9,26)(10,27)(15,28)(16,22)(19,32)(20,21)(29,30)Polycyclic presentation:
2^9|a^2=e,d^2=fi,e^2=g,f^2=h,g^2=i,h^2=i,b^a=bd,d^a=df,f^a=fh,h^a=hi,c^b=cgh,d^b=dfh,e^b=eh,f^b=fhi,g^b=gi,h^b=hi,d^c=di,e^d=ei50 conjugacy classes — view the full character table.
| maximal subgroup of | #72214318, #72214568, #72214592, … |
| maximal normal subgroup of | #72214318, #72214568, #72214592, … |
| maximal quotient of | #72216297, #72225034, C2×#150129 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 57325);