Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #139432 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1898 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C22×C22⋊C4 | C23 |
| prime radical | #15310 | C2 |
| LCS | G ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×#121 ◁ G | |
| LECS | G ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1111 conjugacy classes of subgroups; only the 74 normal subgroups are shown.
This group has 1111 conjugacy classes of subgroups — too many to draw. Only its 74 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)(3,4,8,7)(5,6)(9,10)(11,12,16,13)(14,15);(1,3,7,6,5,8,4,2)(9,11,13,15)(10,14,16,12);(9,12)(10,11)(13,14)(15,16)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,d^2=ghi,b^a=bd,c^a=ce,d^a=dg,e^a=eghi,g^a=gi,h^a=hi,d^b=dh,e^b=eghi,f^b=fhi,g^b=gi,h^b=hi,d^c=dghi,f^c=fghi,f^d=fi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72226008, #72226063, #72226071, … |
| maximal normal subgroup of | #72226008, #72226063, #72226071, … |
| maximal quotient of | #72237831, #72237888, #72285138, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 46668);