Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #485649 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×C4⋊C4 | C24 |
| prime radical | #47177 | C2 |
| LCS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
997 conjugacy classes of subgroups; only the 120 normal subgroups are shown.
This group has 997 conjugacy classes of subgroups — too many to draw. Only its 120 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 — 4 generators on 32 points:
(1,2,6,13)(3,11,15,29)(4,9,18,12)(5,10,17,27)(7,22,8,14)(16,24,31,28)(19,30,20,26)(21,25,32,23);(2,9)(3,15)(4,18)(5,17)(10,23)(11,28)(12,13)(14,26)(19,20)(22,30)(24,29)(25,27);(2,10)(3,5)(4,19)(9,23)(11,14)(12,25)(13,27)(15,17)(16,31)(18,20)(21,32)(22,29)(24,30)(26,28);(1,3,8,5)(2,11,14,10)(4,16,20,21)(6,15,7,17)(9,24,26,25)(12,28,30,23)(13,29,22,27)(18,31,19,32)Polycyclic presentation:
2^9|a^2=g,d^2=i,e^2=g,f^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fi,h^a=hi,d^b=dg,e^b=eg,f^b=fhi,d^c=di,e^c=eh,f^c=fi,f^e=fi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72251747, #72251767 |
| maximal normal subgroup of | #72251747, #72251767 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 392929);