Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #150393 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #11753 |
| commutator | C2×C22⋊C4 | C22×C4 |
| Frattini, cosocle | #379 | C23 |
| prime radical | #23984 | C2 |
| LCS | G ▷ C2×C22⋊C4 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C25 ◁ C2×#379 ◁ G | |
| LECS | G ▷ #379 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
920 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 920 conjugacy classes of subgroups — too many to draw. Only its 54 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) = 16 |
A minimal faithful permutation representation — 3 generators on 16 points:
(1,2)(3,9)(4,12)(5,8)(6,10,14,16)(7,13)(11,15);(1,3)(2,6)(4,10)(5,11)(7,15)(8,14)(9,12)(13,16);(4,13)(7,12)Polycyclic presentation:
2^9|a^2=e,d^2=g,b^a=bd,d^a=df,f^a=fh,g^a=gi,c^b=cf,d^b=dg,e^b=efg,h^b=hi,d^c=dhi,e^d=eh,f^d=fi41 conjugacy classes — view the full character table.
| maximal subgroup of | #72210482, #72210571, #72216452, … |
| maximal normal subgroup of | #72210482, #72216452, #72216457, … |
| maximal quotient of | C2×#150393, #72236933, #72236934, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 57589);