Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #110086 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1437 |
| commutator | C24 | C2×C42 |
| Frattini, cosocle | C26 | C23 |
| prime radical | C22×#120 | C22 |
| LCS | G ▷ C24 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C26 ◁ G | |
| LECS | G ▷ C26 ▷ C22 ▷ C1 | |
1763 conjugacy classes of subgroups; only the 174 normal subgroups are shown.
This group has 1763 conjugacy classes of subgroups — too many to draw. Only its 174 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,13,15,16)(12,14);(1,3)(2,5,6,7)(4,8)(9,11)(10,13,14,16)(12,15);(9,11)(10,13,14,16)(12,15)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=ef,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gi,d^b=di,e^b=eh,f^b=fh,d^c=dh,e^c=eh,f^c=fh74 conjugacy classes — view the full character table.
| maximal subgroup of | #72210578, #72216452, #72216620, … |
| maximal normal subgroup of | #72210578, #72216452, #72216620, … |
| maximal quotient of | #72210444, #72210526, #72213513, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 17282);