Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11522 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #410 |
| commutator | C22×C4 | C2×C42 |
| Frattini, prime radical, cosocle | C22×#41 | C23 |
| socle | C22 | #1799 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C8 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
670 conjugacy classes of subgroups; only the 150 normal subgroups are shown.
This group has 670 conjugacy classes of subgroups — too many to draw. Only its 150 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:
(5,6)(7,18,20,29)(8,21,23,12)(9,14)(10,15,26,33)(11,17)(13,22)(16,25)(19,32)(24,30,36,27)(28,35)(31,34);(1,2,3,4)(5,7,9,8,11,20,25,23)(6,12,16,29,17,21,14,18)(10,19,24,22,26,34,36,35)(13,30,32,15,28,27,31,33);(5,8,9,20,11,23,25,7)(6,13,14,31,17,28,16,32)(10,22,24,34,26,35,36,19)(12,27,18,15,21,30,29,33)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=d,d^2=i,g^2=i,b^a=bd,c^a=ce,e^a=eh,g^a=gi,h^a=hi,c^b=ci,f^b=fi,f^c=fh,f^e=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72311867, #72311876, #72311877, … |
| maximal normal subgroup of | #72311867, #72311876, #72311877, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54057);