Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #63011 |
| automorphism | ? |
| homology | Abelianization (H1) = C27 · Schur multiplier (H2) = C219 |
| direct product | C2×#63040 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C26 |
| commutator, Frattini, cosocle | C22 | C27 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
40620 conjugacy classes of subgroups; only the 29370 normal subgroups are shown.
This group has 40620 conjugacy classes of subgroups — too many to draw. Only its 29370 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 7 generators on 18 points:
(1,2);(3,4,8,7)(5,10,6,9)(11,12,16,15)(13,18,14,17);(3,4,8,7)(5,10,6,9)(11,13,16,14)(12,17,15,18);(3,5,8,6)(4,9,7,10)(11,13,16,14)(12,17,15,18);(11,13,16,14)(12,17,15,18);(3,6,8,5)(4,9,7,10)(11,14,16,13)(12,17,15,18);(3,7,8,4)(5,10,6,9)(11,15,16,12)(13,18,14,17)152 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#511838, #72261011, #72261049, … |
| maximal normal subgroup of | C2×#511838, #72261011, #72261049, … |
| maximal quotient of | C2×#7688517, C2×#8195113, C2×#7880291, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10493560);