Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #111931 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1438 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C2×#2815 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
1318 conjugacy classes of subgroups; only the 134 normal subgroups are shown.
This group has 1318 conjugacy classes of subgroups — too many to draw. Only its 134 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)(3,13,16,29)(5,8,19,25)(6,12,20,30)(7,22,24,15)(9,26,27,17)(10,18,28,21)(14,23,31,32);(2,7)(3,14)(4,18)(8,23)(9,12)(10,13)(11,29)(15,17)(16,26)(19,20)(22,31)(24,28)(25,27)(30,32);(2,8)(3,15)(4,19)(7,9)(10,12)(13,23)(14,17)(16,31)(18,20)(22,26)(24,29)(27,32)Polycyclic presentation:
2^9|a^2=g,d^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fhi,c^b=cf,d^b=di,g^b=ghi,d^c=dhi,g^c=gh50 conjugacy classes — view the full character table.
| maximal subgroup of | #72233123, #72233125, #72233134, … |
| maximal normal subgroup of | #72233123, #72233125, #72233134, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 19127);