Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #98566 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #379 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C22×C42 | C23 |
| prime radical | C24 | #120 |
| LCS | G ▷ C2×C42 ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C23 ▷ C1 | |
634 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 634 conjugacy classes of subgroups — too many to draw. Only its 94 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) = 128 |
A minimal faithful permutation representation — 3 generators on 32 points:
(2,6,7,8)(9,10,13,15,12,11,16,14)(17,18,22,24,21,20,28,29)(19,23,27,31,26,25,30,32);(1,2,5,7)(3,6,4,8)(13,16)(14,15)(17,19,21,26)(18,23,20,25)(22,30,28,27)(24,32,29,31);(1,3,5,4)(2,6,7,8)(9,11)(10,12)(13,14)(15,16)(17,20)(18,21)(19,25)(22,24)(23,26)(27,31)(28,29)(30,32)Polycyclic presentation:
2^9|a^2=f,b^2=ghi,c^2=h,d^2=h,e^2=g,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=ei,d^b=dh,e^b=eg,f^b=fgh,d^c=dg,e^c=eg,f^c=fgi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72238884, #72238885, #72238887, … |
| maximal normal subgroup of | #72238884, #72238885, #72238887, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5813);