Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12096 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | B(4,𝔽2) |
| commutator | C2×C4⋊C4 | C22×C4 |
| Frattini, cosocle | C22×C4⋊C4 | C23 |
| prime radical | C2×#2006 | C2 |
| LCS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24 ◁ C22×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1019 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 1019 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) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 20 points:
(2,4)(5,6,7,10,8,11,12,9)(13,14)(15,18,20,17)(16,19);(1,2,3,4)(6,9)(7,12)(10,11)(15,20);(14,17)(15,20)(18,19)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,d^2=g,e^2=i,f^2=g,b^a=bd,c^a=ce,e^a=ehi,h^a=hi,d^b=dg,e^b=ehi,f^b=fg,d^c=dh,e^c=ei,f^c=fh,e^d=ei,f^e=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72240330, #72240573, #72240577, … |
| maximal normal subgroup of | #72240330, #72240573, #72240577, … |
| maximal quotient of | #72261575 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 44331);