Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #122865 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#331 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C82 | C23 |
| prime radical | #19603 | C2 |
| LCS | G ▷ C4×C8 ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C82 ▷ C42 ▷ C22 ▷ C1 | |
304 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 304 conjugacy classes of subgroups — too many to draw. Only its 82 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,6)(4,13)(5,7)(8,10)(9,11)(12,14)(15,16)(17,18,21,25,30,23,32,28,22,26,31,24,20,19,27,29);(2,6)(3,10)(4,11)(7,14)(8,9)(12,16)(13,15)(17,19)(18,20)(21,25)(22,23)(24,31)(26,30)(27,28)(29,32);(2,7)(6,14)(8,15)(9,13)(17,20,22,30)(18,23,26,19)(21,27,31,32)(24,29,25,28)Polycyclic presentation:
2^9|a^2=e,c^2=hi,d^2=gi,e^2=fgh,f^2=h,g^2=i,b^a=bd,c^a=ci,d^a=df,f^a=fh,g^a=gh,d^b=dg,e^b=efgh,f^b=fh,g^b=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72216572, #72225233, #72225245, … |
| maximal normal subgroup of | #72216572, #72225233, #72225245, … |
| maximal quotient of | #72216566 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30106);