Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #105580 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C4×C22⋊C4 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #1749 |
| prime radical | C25 | M16 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
694 conjugacy classes of subgroups; only the 122 normal subgroups are shown.
This group has 694 conjugacy classes of subgroups — too many to draw. Only its 122 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,7)(4,6)(5,8)(10,15,17,16)(18,22,24,23);(1,3)(2,5,6,7)(4,8)(9,10,12,18,14,15,21,22,13,17,20,24,11,16,19,23);(10,16,17,15)(12,20)(18,22,24,23)(19,21)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=e,d^2=i,f^2=h,h^2=i,b^a=bd,f^a=fgi,c^b=ch,d^b=dg,e^b=ei,f^c=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72219601, #72224575, #72239329, … |
| maximal normal subgroup of | #72219601, #72224575, #72239329, … |
| maximal quotient of | C2×#128354 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 35550);