Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #152116 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #12697 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | #532 | C23 |
| prime radical | #60801 | C2 |
| LCS | G ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×C42 ◁ #3444 ◁ G | |
| LECS | G ▷ #532 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
667 conjugacy classes of subgroups; only the 76 normal subgroups are shown.
This group has 667 conjugacy classes of subgroups — too many to draw. Only its 76 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)(3,13,17,24)(4,19,21,11)(5,9,20,29)(6,10)(7,12)(8,15,25,22)(14,28,18,31)(16,27,32,26)(23,30);(1,3)(2,8)(4,14)(5,15)(6,22)(7,18)(9,28)(10,31)(11,25)(12,27)(13,30)(16,23)(17,21)(19,26)(20,32)(24,29);(1,4,7,21)(2,9)(3,14,18,17)(5,6,20,23)(8,24)(10,19)(11,30)(12,29)(13,26)(15,22,32,16)(25,31)(27,28)Polycyclic presentation:
2^9|a^2=f,c^2=hi,d^2=h,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gh,h^a=hi,c^b=ch,d^b=dh,e^b=ei,f^b=fghi,g^b=gi,f^c=fi,f^d=fhi,g^f=gi38 conjugacy classes — view the full character table.
| maximal subgroup of | #72306760, #72306779 |
| maximal normal subgroup of | #72306760, #72306779 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59313);