Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #465 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C24×C42 |
| direct product | C4×#1845 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C2×D4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle, prime radical | C24 | C22×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1630 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,3,4);(6,8)(9,10,11,12)(14,17)(15,18)(19,20);(13,14,18,20,16,19,15,17);(5,6,7,8)176 conjugacy classes — view the full character table.
| maximal subgroup of | #72210318, #72210403, #72210445, … |
| maximal normal subgroup of | #72210318, #72210403, #72210445, … |
| maximal quotient of | #72210308, #72210396, #72210403, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 262176);