Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #7527 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C2×C8 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1408 |
| commutator | C4×C8 | C42 |
| Frattini, cosocle | C2×#492 | C22 |
| prime radical | C2×#127 | D4 |
| LCS | G ▷ C4×C8 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ C2×C42 ◁ C2×#492 ◁ G | |
| LECS | G ▷ C2×#492 ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
229 conjugacy classes of subgroups; only the 38 normal subgroups are shown.
This group has 229 conjugacy classes of subgroups — too many to draw. Only its 38 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) = 64 |
A minimal faithful permutation representation — 2 generators on 20 points:
(5,6)(7,14,17,10)(8,19,16,12)(9,13)(11,18,20,15);(1,2,3,4)(6,10,12,11,13,20,19,14)(7,15)(17,18)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=ghi,e^2=h,f^2=hi,g^2=i,b^a=bc,c^a=cf,e^a=egh,f^a=fg,g^a=gh,h^a=hi,c^b=ci,d^b=dfgi,d^c=dgh,e^d=eh,f^d=fhi,g^d=gi38 conjugacy classes — view the full character table.
| maximal subgroup of | #72309199, #72309258, #72309259 |
| maximal normal subgroup of | #72309199, #72309258, #72309259 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2000);