Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2199 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | B(4,𝔽2) |
| commutator | C2×D4 | C2×C42 |
| Frattini, prime radical, cosocle | C22×#41 | C23 |
| socle | C22 | #1889 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
693 conjugacy classes of subgroups; only the 154 normal subgroups are shown.
This group has 693 conjugacy classes of subgroups — too many to draw. Only its 154 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2,4,3)(5,6)(7,17)(8,14)(9,24)(10,15)(11,16)(12,19)(13,22)(18,29)(20,30)(21,33)(23,31)(25,27)(26,32)(28,34)(35,36);(5,7,10,20,11,21,9,25)(6,12,15,28,16,29,24,36)(8,19,22,34,23,18,26,35)(13,30,31,33,32,27,14,17);(1,3,4,2)(5,8)(6,13)(7,18)(9,26)(10,22)(11,23)(12,27)(14,24)(15,31)(16,32)(17,28)(19,21)(20,35)(25,34)(29,30)(33,36)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=f,e^2=i,g^2=i,b^a=bd,c^a=ce,e^a=ei,h^a=hi,c^b=ci,e^b=ehi,d^c=dh,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72312470, #72312494, #72312521, … |
| maximal normal subgroup of | #72312470, #72312494, #72312521, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54536);